Concentration of measure and mixing for Markov chains Malwina J - - PowerPoint PPT Presentation

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure and mixing for Markov chains

Malwina J Luczak

Department of Mathematics London School of Economics London WC2A 2AE UK e-mail: m.j.luczak@lse.ac.uk

June 2009

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Complex random networks

◮ Many complex systems (e.g. Internet, biological networks,

communications networks) can be modelled by Markov chains

◮ Under suitable conditions there is a law of large numbers, i.e.

random system close to a deterministic process with simpler dynamics, derived from average drift.

◮ Such laws of large numbers and quantitative concentration of

measure estimates are often easier to establish over a bounded time interval, starting from a fixed state with some nice properties.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Complex random networks

◮ Many complex systems (e.g. Internet, biological networks,

communications networks) can be modelled by Markov chains

◮ Under suitable conditions there is a law of large numbers, i.e.

random system close to a deterministic process with simpler dynamics, derived from average drift.

◮ Such laws of large numbers and quantitative concentration of

measure estimates are often easier to establish over a bounded time interval, starting from a fixed state with some nice properties.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Complex random networks

◮ Many complex systems (e.g. Internet, biological networks,

communications networks) can be modelled by Markov chains

◮ Under suitable conditions there is a law of large numbers, i.e.

random system close to a deterministic process with simpler dynamics, derived from average drift.

◮ Such laws of large numbers and quantitative concentration of

measure estimates are often easier to establish over a bounded time interval, starting from a fixed state with some nice properties.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Complex random networks

◮ Can we use time-dependent laws of large numbers and

concentration to deduce a law of large numbers and concentration of measure in equilibrium?

◮ That is, can we show that the equilibrium behaviour of the

Markov process resembles equilibrium behaviour of the approximating deterministic system?

◮ Not in general, but sometimes yes.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Complex random networks

◮ Can we use time-dependent laws of large numbers and

concentration to deduce a law of large numbers and concentration of measure in equilibrium?

◮ That is, can we show that the equilibrium behaviour of the

Markov process resembles equilibrium behaviour of the approximating deterministic system?

◮ Not in general, but sometimes yes.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Complex random networks

◮ Can we use time-dependent laws of large numbers and

concentration to deduce a law of large numbers and concentration of measure in equilibrium?

◮ That is, can we show that the equilibrium behaviour of the

Markov process resembles equilibrium behaviour of the approximating deterministic system?

◮ Not in general, but sometimes yes.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure in equilibrium and mixing

◮ Many systems can be viewed as interacting systems of

particles.

◮ For instance, these can be nodes with spins, vertices with

colours, or servers in queueing network.

◮ Often in such systems concentration of measure in stationary

distribution co-occurs with rapid mixing of the Markov chain.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure in equilibrium and mixing

◮ Many systems can be viewed as interacting systems of

particles.

◮ For instance, these can be nodes with spins, vertices with

colours, or servers in queueing network.

◮ Often in such systems concentration of measure in stationary

distribution co-occurs with rapid mixing of the Markov chain.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure in equilibrium and mixing

◮ Many systems can be viewed as interacting systems of

particles.

◮ For instance, these can be nodes with spins, vertices with

colours, or servers in queueing network.

◮ Often in such systems concentration of measure in stationary

distribution co-occurs with rapid mixing of the Markov chain.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure in equilibrium and mixing

◮ Here, by concentration of measure we mean that, for nice

functions f on the state space, the probability that f deviates from its equilibrium mean by more than u decays like e−u2/cn, where n is the order of E[f (Xt)] and, usually, a measure of system size.

◮ Rapid mixing means convergence to equilibrium in a time of

the order O(n log n) steps, not just polynomial in n for a system of size n.

◮ A notable example is the Glauber dynamics for the mean-field

Ising model, discussed in more detail shortly.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure in equilibrium and mixing

◮ Here, by concentration of measure we mean that, for nice

functions f on the state space, the probability that f deviates from its equilibrium mean by more than u decays like e−u2/cn, where n is the order of E[f (Xt)] and, usually, a measure of system size.

◮ Rapid mixing means convergence to equilibrium in a time of

the order O(n log n) steps, not just polynomial in n for a system of size n.

◮ A notable example is the Glauber dynamics for the mean-field

Ising model, discussed in more detail shortly.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Concentration of measure in equilibrium and mixing

◮ Here, by concentration of measure we mean that, for nice

functions f on the state space, the probability that f deviates from its equilibrium mean by more than u decays like e−u2/cn, where n is the order of E[f (Xt)] and, usually, a measure of system size.

◮ Rapid mixing means convergence to equilibrium in a time of

the order O(n log n) steps, not just polynomial in n for a system of size n.

◮ A notable example is the Glauber dynamics for the mean-field

Ising model, discussed in more detail shortly.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Other related phenomena

◮ Also related is the cut-off phenomenon, i.e., a sharp threshold

for mixing.

◮ Cut-offs have been proven for chains making local moves, like

Glauber dynamics, and rapidly mixing.

◮ Also, a rapidly mixing and concentrated Markov chain

representing the evolution of an interacting particle system, will often exhibit chaoticity, i.e. approximate asymptotic independence of particles as the system grows large.

◮ We aim to discuss some ideas relating concentration, rapid

mixing, chaoticity and cut-offs to one another. First, an example.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Other related phenomena

◮ Also related is the cut-off phenomenon, i.e., a sharp threshold

for mixing.

◮ Cut-offs have been proven for chains making local moves, like

Glauber dynamics, and rapidly mixing.

◮ Also, a rapidly mixing and concentrated Markov chain

representing the evolution of an interacting particle system, will often exhibit chaoticity, i.e. approximate asymptotic independence of particles as the system grows large.

◮ We aim to discuss some ideas relating concentration, rapid

mixing, chaoticity and cut-offs to one another. First, an example.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Other related phenomena

◮ Also related is the cut-off phenomenon, i.e., a sharp threshold

for mixing.

◮ Cut-offs have been proven for chains making local moves, like

Glauber dynamics, and rapidly mixing.

◮ Also, a rapidly mixing and concentrated Markov chain

representing the evolution of an interacting particle system, will often exhibit chaoticity, i.e. approximate asymptotic independence of particles as the system grows large.

◮ We aim to discuss some ideas relating concentration, rapid

mixing, chaoticity and cut-offs to one another. First, an example.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Other related phenomena

◮ Also related is the cut-off phenomenon, i.e., a sharp threshold

for mixing.

◮ Cut-offs have been proven for chains making local moves, like

Glauber dynamics, and rapidly mixing.

◮ Also, a rapidly mixing and concentrated Markov chain

representing the evolution of an interacting particle system, will often exhibit chaoticity, i.e. approximate asymptotic independence of particles as the system grows large.

◮ We aim to discuss some ideas relating concentration, rapid

mixing, chaoticity and cut-offs to one another. First, an example.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ Let G = (V , E) be a finite graph. ◮ Configurations are elements of S := {−1, 1}V , and for σ ∈ S,

the spin at v is σ(v).

◮ The energy H(σ) of σ ∈ {−1, 1}V is

H(σ) := −J

• v,w∈V ,

v∼w

σ(v)σ(w), (1)

◮ The quantity J > 0 measures interaction strength between

vertices.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ Let G = (V , E) be a finite graph. ◮ Configurations are elements of S := {−1, 1}V , and for σ ∈ S,

the spin at v is σ(v).

◮ The energy H(σ) of σ ∈ {−1, 1}V is

H(σ) := −J

• v,w∈V ,

v∼w

σ(v)σ(w), (1)

◮ The quantity J > 0 measures interaction strength between

vertices.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ Let G = (V , E) be a finite graph. ◮ Configurations are elements of S := {−1, 1}V , and for σ ∈ S,

the spin at v is σ(v).

◮ The energy H(σ) of σ ∈ {−1, 1}V is

H(σ) := −J

• v,w∈V ,

v∼w

σ(v)σ(w), (1)

◮ The quantity J > 0 measures interaction strength between

vertices.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ Let G = (V , E) be a finite graph. ◮ Configurations are elements of S := {−1, 1}V , and for σ ∈ S,

the spin at v is σ(v).

◮ The energy H(σ) of σ ∈ {−1, 1}V is

H(σ) := −J

• v,w∈V ,

v∼w

σ(v)σ(w), (1)

◮ The quantity J > 0 measures interaction strength between

vertices.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ The Gibbs measure on G with parameter β ≥ 0 is given by

π(σ) = e−βH(σ) Z(β) . (2)

◮ Here Z(β) = σ∈S e−βH(σ) is a normalising constant. ◮ β is the inverse of temperature, and measures the influence of

the energy H on the Gibbs measure.

◮ At infinite temperature (β = 0), the measure π is uniform

• ver S and the random variables {σ(v)}v∈V are independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ The Gibbs measure on G with parameter β ≥ 0 is given by

π(σ) = e−βH(σ) Z(β) . (2)

◮ Here Z(β) = σ∈S e−βH(σ) is a normalising constant. ◮ β is the inverse of temperature, and measures the influence of

the energy H on the Gibbs measure.

◮ At infinite temperature (β = 0), the measure π is uniform

• ver S and the random variables {σ(v)}v∈V are independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ The Gibbs measure on G with parameter β ≥ 0 is given by

π(σ) = e−βH(σ) Z(β) . (2)

◮ Here Z(β) = σ∈S e−βH(σ) is a normalising constant. ◮ β is the inverse of temperature, and measures the influence of

the energy H on the Gibbs measure.

◮ At infinite temperature (β = 0), the measure π is uniform

• ver S and the random variables {σ(v)}v∈V are independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field Ising model

◮ The Gibbs measure on G with parameter β ≥ 0 is given by

π(σ) = e−βH(σ) Z(β) . (2)

◮ Here Z(β) = σ∈S e−βH(σ) is a normalising constant. ◮ β is the inverse of temperature, and measures the influence of

the energy H on the Gibbs measure.

◮ At infinite temperature (β = 0), the measure π is uniform

• ver S and the random variables {σ(v)}v∈V are independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ The (single-site) Glauber dynamics for π is the Markov chain

(Xt) on S with transitions as follows.

◮ When at σ, choose vertex v is u.a.r., and generate a new state

from π conditioned on {η ∈ Ω : η(w) = σ(w), w = v}.

◮ The new state will agree with σ except maybe at v, where the

new spin is +1 with probability p(σ; v) := eβMv(σ) eβMv(σ) + e−βMv(σ) , (3) with Mv(σ) := J

w : w∼v σ(w).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ The (single-site) Glauber dynamics for π is the Markov chain

(Xt) on S with transitions as follows.

◮ When at σ, choose vertex v is u.a.r., and generate a new state

from π conditioned on {η ∈ Ω : η(w) = σ(w), w = v}.

◮ The new state will agree with σ except maybe at v, where the

new spin is +1 with probability p(σ; v) := eβMv(σ) eβMv(σ) + e−βMv(σ) , (3) with Mv(σ) := J

w : w∼v σ(w).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ The (single-site) Glauber dynamics for π is the Markov chain

(Xt) on S with transitions as follows.

◮ When at σ, choose vertex v is u.a.r., and generate a new state

from π conditioned on {η ∈ Ω : η(w) = σ(w), w = v}.

◮ The new state will agree with σ except maybe at v, where the

new spin is +1 with probability p(σ; v) := eβMv(σ) eβMv(σ) + e−βMv(σ) , (3) with Mv(σ) := J

w : w∼v σ(w).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ Evidently, the distribution of the new spin at v depends only

• n the current spins at the neighbours of v.

◮ (Xt) is reversible with respect to Gibbs measure π, which is

thus its stationary measure.

◮ Given a sequence Gn = (Vn, En) of graphs, let πn be the Ising

measure and (X (n)

t

) the Glauber dynamics on Gn.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ Evidently, the distribution of the new spin at v depends only

• n the current spins at the neighbours of v.

◮ (Xt) is reversible with respect to Gibbs measure π, which is

thus its stationary measure.

◮ Given a sequence Gn = (Vn, En) of graphs, let πn be the Ising

measure and (X (n)

t

) the Glauber dynamics on Gn.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ Evidently, the distribution of the new spin at v depends only

• n the current spins at the neighbours of v.

◮ (Xt) is reversible with respect to Gibbs measure π, which is

thus its stationary measure.

◮ Given a sequence Gn = (Vn, En) of graphs, let πn be the Ising

measure and (X (n)

t

) the Glauber dynamics on Gn.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ For σ ∈ Sn, let L(X (n) t

, σ) be the law of X (n)

t

starting from σ.

◮ The worst-case distance to stationarity of the Glauber

dynamics chain after t steps is dn(t) := max

σ∈Sn dTV(L(X (n) t

, σ), πn). (4)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ For σ ∈ Sn, let L(X (n) t

, σ) be the law of X (n)

t

starting from σ.

◮ The worst-case distance to stationarity of the Glauber

dynamics chain after t steps is dn(t) := max

σ∈Sn dTV(L(X (n) t

, σ), πn). (4)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ The mixing time tmix(n) is defined as

tmix(n) := min{t : dn(t) ≤ 1/4}. (5)

◮ Note tmix(n) is finite for each fixed n since, by the ergodic

theorem for Markov chains, dn(t) → 0 as t → ∞.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ The mixing time tmix(n) is defined as

tmix(n) := min{t : dn(t) ≤ 1/4}. (5)

◮ Note tmix(n) is finite for each fixed n since, by the ergodic

theorem for Markov chains, dn(t) → 0 as t → ∞.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ But tmix(n) will in general tend to infinity with n, and it is

natural to ask about the growth rate of tmix(n).

Definition

The Glauber dynamics is said to exhibit a cut-off at {tn} with window size {wn} if wn = o(tn) and lim

γ→∞ lim inf n→∞ dn(tn − γwn) = 1,

lim

γ→∞ lim sup n→∞ dn(tn + γwn) = 0.

Informally, a cut-off is a sharp threshold for mixing.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ But tmix(n) will in general tend to infinity with n, and it is

natural to ask about the growth rate of tmix(n).

Definition

The Glauber dynamics is said to exhibit a cut-off at {tn} with window size {wn} if wn = o(tn) and lim

γ→∞ lim inf n→∞ dn(tn − γwn) = 1,

lim

γ→∞ lim sup n→∞ dn(tn + γwn) = 0.

Informally, a cut-off is a sharp threshold for mixing.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Glauber dynamics for Ising model

◮ But tmix(n) will in general tend to infinity with n, and it is

natural to ask about the growth rate of tmix(n).

Definition

The Glauber dynamics is said to exhibit a cut-off at {tn} with window size {wn} if wn = o(tn) and lim

γ→∞ lim inf n→∞ dn(tn − γwn) = 1,

lim

γ→∞ lim sup n→∞ dn(tn + γwn) = 0.

Informally, a cut-off is a sharp threshold for mixing.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case

◮ Here we consider the mean-field case, i.e. Gn is Kn, the

complete graph on n vertices.

◮ Take J = 1/n; so measure π on {−1, 1}n is

π(σ) = πn(σ) = 1 Z(β) exp  β n

• 1≤i<j≤n

σ(i)σ(j)   . (6)

◮ We update to a +1 with prob. p+(S(σ) − n−1σ(i)), where

p+(s) = eβs eβs + e−βs .

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case

◮ Here we consider the mean-field case, i.e. Gn is Kn, the

complete graph on n vertices.

◮ Take J = 1/n; so measure π on {−1, 1}n is

π(σ) = πn(σ) = 1 Z(β) exp  β n

• 1≤i<j≤n

σ(i)σ(j)   . (6)

◮ We update to a +1 with prob. p+(S(σ) − n−1σ(i)), where

p+(s) = eβs eβs + e−βs .

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case

◮ Here we consider the mean-field case, i.e. Gn is Kn, the

complete graph on n vertices.

◮ Take J = 1/n; so measure π on {−1, 1}n is

π(σ) = πn(σ) = 1 Z(β) exp  β n

• 1≤i<j≤n

σ(i)σ(j)   . (6)

◮ We update to a +1 with prob. p+(S(σ) − n−1σ(i)), where

p+(s) = eβs eβs + e−βs .

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β < 1

◮ It follows e.g. by a coupling argument, or from the

Dobrushin-Shlosman uniqueness criterion that tmix(n) = O(n log n) when β < 1;

◮ Consider the Hamming distance on S: that is, two vectors in

{−1, 1}n are adjacent if they differ in exactly one co-ordinate (equivalent to ℓ1 distance in this setting).

◮ The Gibbs measure π (stationary measure of GD) exhibits

normal concentration of measure for Lipschitz functions in the following sense.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β < 1

◮ It follows e.g. by a coupling argument, or from the

Dobrushin-Shlosman uniqueness criterion that tmix(n) = O(n log n) when β < 1;

◮ Consider the Hamming distance on S: that is, two vectors in

{−1, 1}n are adjacent if they differ in exactly one co-ordinate (equivalent to ℓ1 distance in this setting).

◮ The Gibbs measure π (stationary measure of GD) exhibits

normal concentration of measure for Lipschitz functions in the following sense.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β < 1

◮ It follows e.g. by a coupling argument, or from the

Dobrushin-Shlosman uniqueness criterion that tmix(n) = O(n log n) when β < 1;

◮ Consider the Hamming distance on S: that is, two vectors in

{−1, 1}n are adjacent if they differ in exactly one co-ordinate (equivalent to ℓ1 distance in this setting).

◮ The Gibbs measure π (stationary measure of GD) exhibits

normal concentration of measure for Lipschitz functions in the following sense.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β < 1

◮ Let X (n) be X (n) t

in equilibrium.

◮ Then, for some c, C > 0,

Pπ(|f (X (n)) − Eπ f (X (n))| ≥ u) ≤ Ce−u2/cn, (7) uniformly over all 1-Lipschitz functions on S and over all n.

◮ Treating (7) as a statement about π without mention of X (n) t

, we can rewrite π({σ : |f (σ) − π(f )| ≥ u}) ≤ Ce−u2/cn.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β < 1

◮ Let X (n) be X (n) t

in equilibrium.

◮ Then, for some c, C > 0,

Pπ(|f (X (n)) − Eπ f (X (n))| ≥ u) ≤ Ce−u2/cn, (7) uniformly over all 1-Lipschitz functions on S and over all n.

◮ Treating (7) as a statement about π without mention of X (n) t

, we can rewrite π({σ : |f (σ) − π(f )| ≥ u}) ≤ Ce−u2/cn.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β < 1

◮ Let X (n) be X (n) t

in equilibrium.

◮ Then, for some c, C > 0,

Pπ(|f (X (n)) − Eπ f (X (n))| ≥ u) ≤ Ce−u2/cn, (7) uniformly over all 1-Lipschitz functions on S and over all n.

◮ Treating (7) as a statement about π without mention of X (n) t

, we can rewrite π({σ : |f (σ) − π(f )| ≥ u}) ≤ Ce−u2/cn.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case: β < 1

◮ Moreover, there is a cut-off.

Theorem

[Levin, L. and Peres (2008)] Suppose that β < 1. The Glauber dynamics for the Ising model on Kn has a cut-off at tn = [2(1 − β)]−1n log n with window size n.

◮ Furthermore, asymptotically the spin values in a bounded set

• f vertices are almost independent (decay of correlations).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case: β < 1

◮ Moreover, there is a cut-off.

Theorem

[Levin, L. and Peres (2008)] Suppose that β < 1. The Glauber dynamics for the Ising model on Kn has a cut-off at tn = [2(1 − β)]−1n log n with window size n.

◮ Furthermore, asymptotically the spin values in a bounded set

• f vertices are almost independent (decay of correlations).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case: β < 1

◮ Moreover, there is a cut-off.

Theorem

[Levin, L. and Peres (2008)] Suppose that β < 1. The Glauber dynamics for the Ising model on Kn has a cut-off at tn = [2(1 − β)]−1n log n with window size n.

◮ Furthermore, asymptotically the spin values in a bounded set

• f vertices are almost independent (decay of correlations).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field model: case β = 1

◮ There is no rapid mixing though tmix(n) is polynomial in n.

Theorem

[Levin, L. and Peres (2008)] Let β = 1. There are C1, C2 > 0 such that C1n3/2 ≤ tmix(n) ≤ C2n3/2.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field model: case β = 1

◮ There is no rapid mixing though tmix(n) is polynomial in n.

Theorem

[Levin, L. and Peres (2008)] Let β = 1. There are C1, C2 > 0 such that C1n3/2 ≤ tmix(n) ≤ C2n3/2.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field model: case β = 1

◮ Also there is no cut-off (Ding, Lubetzky and Peres (2009+)). ◮ And furthermore, there is no meaningful normal concentration

• f measure bound uniform over all 1-Lipschitz functions: for

example, consider m/2, where m(σ) = n

i=1 σ(i). ◮ The probability that function m/n equals x ∈ [−1, 1] is

proportional to exp(−ng(x)), where g(x) has a unique minimum at 0, but the second derivative vanishes.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field model: case β = 1

◮ Also there is no cut-off (Ding, Lubetzky and Peres (2009+)). ◮ And furthermore, there is no meaningful normal concentration

• f measure bound uniform over all 1-Lipschitz functions: for

example, consider m/2, where m(σ) = n

i=1 σ(i). ◮ The probability that function m/n equals x ∈ [−1, 1] is

proportional to exp(−ng(x)), where g(x) has a unique minimum at 0, but the second derivative vanishes.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field model: case β = 1

◮ Also there is no cut-off (Ding, Lubetzky and Peres (2009+)). ◮ And furthermore, there is no meaningful normal concentration

• f measure bound uniform over all 1-Lipschitz functions: for

example, consider m/2, where m(σ) = n

i=1 σ(i). ◮ The probability that function m/n equals x ∈ [−1, 1] is

proportional to exp(−ng(x)), where g(x) has a unique minimum at 0, but the second derivative vanishes.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β > 1

◮ Here tmix(n) is exponential in n. This can be proved e.g. by

upper bounding the Cheeger constant Φ = min

A:π(A)≤1/2

• σ∈A,τ∈A π(σ)P(σ, τ)

π(A) .

◮ Also m(X (n)) is bi-modal: for some c > 0,

π({σ : m(σ) ≥ cn}) = π({σ : m(σ) ≤ −cn}) ≥ 1/4.

◮ Also, for β ≥ 1 the spins are far from independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β > 1

◮ Here tmix(n) is exponential in n. This can be proved e.g. by

upper bounding the Cheeger constant Φ = min

A:π(A)≤1/2

• σ∈A,τ∈A π(σ)P(σ, τ)

π(A) .

◮ Also m(X (n)) is bi-modal: for some c > 0,

π({σ : m(σ) ≥ cn}) = π({σ : m(σ) ≤ −cn}) ≥ 1/4.

◮ Also, for β ≥ 1 the spins are far from independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Mean-field case β > 1

◮ Here tmix(n) is exponential in n. This can be proved e.g. by

upper bounding the Cheeger constant Φ = min

A:π(A)≤1/2

• σ∈A,τ∈A π(σ)P(σ, τ)

π(A) .

◮ Also m(X (n)) is bi-modal: for some c > 0,

π({σ : m(σ) ≥ cn}) = π({σ : m(σ) ≤ −cn}) ≥ 1/4.

◮ Also, for β ≥ 1 the spins are far from independent.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Our general setting

◮ X = (Xt)t∈Z+ is a discrete-time Markov chain with discrete

state space S.

◮ Transition probabilities are P(x, y) for x, y ∈ S. ◮ For every x, y ∈ S, P(x, y) > 0 if and only if P(y, x) > 0. ◮ Form an undirected graph with vertex set S where {x, y} is

an edge if and only if P(x, y) > 0 and x = y. The graph is locally finite, i.e. each vertex has finitely many neighbours.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Our general setting

◮ X = (Xt)t∈Z+ is a discrete-time Markov chain with discrete

state space S.

◮ Transition probabilities are P(x, y) for x, y ∈ S. ◮ For every x, y ∈ S, P(x, y) > 0 if and only if P(y, x) > 0. ◮ Form an undirected graph with vertex set S where {x, y} is

an edge if and only if P(x, y) > 0 and x = y. The graph is locally finite, i.e. each vertex has finitely many neighbours.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Our general setting

◮ X = (Xt)t∈Z+ is a discrete-time Markov chain with discrete

state space S.

◮ Transition probabilities are P(x, y) for x, y ∈ S. ◮ For every x, y ∈ S, P(x, y) > 0 if and only if P(y, x) > 0. ◮ Form an undirected graph with vertex set S where {x, y} is

an edge if and only if P(x, y) > 0 and x = y. The graph is locally finite, i.e. each vertex has finitely many neighbours.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Our general setting

◮ X = (Xt)t∈Z+ is a discrete-time Markov chain with discrete

state space S.

◮ Transition probabilities are P(x, y) for x, y ∈ S. ◮ For every x, y ∈ S, P(x, y) > 0 if and only if P(y, x) > 0. ◮ Form an undirected graph with vertex set S where {x, y} is

an edge if and only if P(x, y) > 0 and x = y. The graph is locally finite, i.e. each vertex has finitely many neighbours.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

General setting continued

◮ Xt may be lazy, i.e. can have P(x, x) > 0 for some x ∈ S. ◮ Endow S with a graph metric d given by d(x, y) = 1 if

P(x, y) > 0 and x = y, and for non-adjacent x, y, d(x, y) the length of the shortest path between x and y in the graph.

◮ Assume the graph is connected. ◮ Very natural setting, and many models in applied probability

and combinatorics fit into this framework, including the Glauber dynamics for Ising model.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

General setting continued

◮ Xt may be lazy, i.e. can have P(x, x) > 0 for some x ∈ S. ◮ Endow S with a graph metric d given by d(x, y) = 1 if

P(x, y) > 0 and x = y, and for non-adjacent x, y, d(x, y) the length of the shortest path between x and y in the graph.

◮ Assume the graph is connected. ◮ Very natural setting, and many models in applied probability

and combinatorics fit into this framework, including the Glauber dynamics for Ising model.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

General setting continued

◮ Xt may be lazy, i.e. can have P(x, x) > 0 for some x ∈ S. ◮ Endow S with a graph metric d given by d(x, y) = 1 if

P(x, y) > 0 and x = y, and for non-adjacent x, y, d(x, y) the length of the shortest path between x and y in the graph.

◮ Assume the graph is connected. ◮ Very natural setting, and many models in applied probability

and combinatorics fit into this framework, including the Glauber dynamics for Ising model.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

General setting continued

◮ Xt may be lazy, i.e. can have P(x, x) > 0 for some x ∈ S. ◮ Endow S with a graph metric d given by d(x, y) = 1 if

P(x, y) > 0 and x = y, and for non-adjacent x, y, d(x, y) the length of the shortest path between x and y in the graph.

◮ Assume the graph is connected. ◮ Very natural setting, and many models in applied probability

and combinatorics fit into this framework, including the Glauber dynamics for Ising model.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Underlying probability space

◮ Each Xt can be seen as a r.v. on (Ω, F), where

Ω = {ω = (ω0, ω1, . . .) : ωi ∈ S ∀i}, and Xt is the t-co-ordinate projection, i.e. Xt(ω) = ωt.

◮ Also, F = σ(∪∞ t=0Ft), with Ft = σ(Xi : i ≤ t). ◮ Law of (Xt) can be viewed as a probability measure P on

(Ω, F): P({ω : ωj = xj : j ≤ t}) = µ({x0})

t−1

• j=0

P(xj, xj+1),

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Underlying probability space

◮ Each Xt can be seen as a r.v. on (Ω, F), where

Ω = {ω = (ω0, ω1, . . .) : ωi ∈ S ∀i}, and Xt is the t-co-ordinate projection, i.e. Xt(ω) = ωt.

◮ Also, F = σ(∪∞ t=0Ft), with Ft = σ(Xi : i ≤ t). ◮ Law of (Xt) can be viewed as a probability measure P on

(Ω, F): P({ω : ωj = xj : j ≤ t}) = µ({x0})

t−1

• j=0

P(xj, xj+1),

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Underlying probability space

◮ Each Xt can be seen as a r.v. on (Ω, F), where

Ω = {ω = (ω0, ω1, . . .) : ωi ∈ S ∀i}, and Xt is the t-co-ordinate projection, i.e. Xt(ω) = ωt.

◮ Also, F = σ(∪∞ t=0Ft), with Ft = σ(Xi : i ≤ t). ◮ Law of (Xt) can be viewed as a probability measure P on

(Ω, F): P({ω : ωj = xj : j ≤ t}) = µ({x0})

t−1

• j=0

P(xj, xj+1),

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More definitions

◮ This is Pµ, law of (Xt) conditional on L(X0) = µ. ◮ Pt(x, y) is the t-step transition probability from x to y, given

inductively by Pt(x, y) =

• z∈S

Pt−1(x, z)P(z, y).

◮ Then Pµ(Xt ∈ A) = (µPt)(A) for A ⊆ S.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More definitions

◮ This is Pµ, law of (Xt) conditional on L(X0) = µ. ◮ Pt(x, y) is the t-step transition probability from x to y, given

inductively by Pt(x, y) =

• z∈S

Pt−1(x, z)P(z, y).

◮ Then Pµ(Xt ∈ A) = (µPt)(A) for A ⊆ S.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More definitions

◮ This is Pµ, law of (Xt) conditional on L(X0) = µ. ◮ Pt(x, y) is the t-step transition probability from x to y, given

inductively by Pt(x, y) =

• z∈S

Pt−1(x, z)P(z, y).

◮ Then Pµ(Xt ∈ A) = (µPt)(A) for A ⊆ S.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More definitions

◮ Eµ is the corresponding expectation operator. ◮ For t ∈ Z+ and f : S → R,

(Ptf )(x) =

• y

Pt(x, y)f (y), x ∈ S.

◮ So (Ptf )(x) = Eδx[f (Xt)] = (δxPt)(f ), the expected value of

f (Xt) conditional on Xt starting at x.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More definitions

◮ Eµ is the corresponding expectation operator. ◮ For t ∈ Z+ and f : S → R,

(Ptf )(x) =

• y

Pt(x, y)f (y), x ∈ S.

◮ So (Ptf )(x) = Eδx[f (Xt)] = (δxPt)(f ), the expected value of

f (Xt) conditional on Xt starting at x.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More definitions

◮ Eµ is the corresponding expectation operator. ◮ For t ∈ Z+ and f : S → R,

(Ptf )(x) =

• y

Pt(x, y)f (y), x ∈ S.

◮ So (Ptf )(x) = Eδx[f (Xt)] = (δxPt)(f ), the expected value of

f (Xt) conditional on Xt starting at x.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Lipschitz functions

◮ Function f : S → R is Lipschitz (or 1-Lipschtitz) if

f Lip= sup

x=y

|f (x) − f (y)| d(x, y) ≤ 1.

◮ Equivalently, f is Lipschitz if supx,y:d(x,y)=1 |f (x) − f (y)| ≤ 1.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Lipschitz functions

◮ Function f : S → R is Lipschitz (or 1-Lipschtitz) if

f Lip= sup

x=y

|f (x) − f (y)| d(x, y) ≤ 1.

◮ Equivalently, f is Lipschitz if supx,y:d(x,y)=1 |f (x) − f (y)| ≤ 1.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Concentration for Lipschitz functions

◮ For a prob measure µ on (S, P(S)) and a r.v. X with law

L(X) = µ, µ or X has normal concentration if for all u > 0, uniformly over 1-Lipschitz functions f , µ(|f (X) − µ(f )| ≥ u) ≤ Ce−cu2. (8)

◮ We shall give conditions when (Xt) has normal concentration

• f measure long term and in equilibrium.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Concentration for Lipschitz functions

◮ For a prob measure µ on (S, P(S)) and a r.v. X with law

L(X) = µ, µ or X has normal concentration if for all u > 0, uniformly over 1-Lipschitz functions f , µ(|f (X) − µ(f )| ≥ u) ≤ Ce−cu2. (8)

◮ We shall give conditions when (Xt) has normal concentration

• f measure long term and in equilibrium.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Wasserstein distance

Wasserstein distance between measures µ1 and µ2 is dW (µ1, µ2) = sup

f

• fdµ1 −
• fdµ2
• = sup

f

|µ1(f ) − µ2(f )|, where the sup is over measurable 1-Lipschitz functions f : S → R. By the Kantorovich – Rubinstein theorem dW (µ1, µ2) = inf

π {π[d(X, Y )] : L(X) = µ1, L(Y ) = µ2},

with inf over all couplings π on S × S with marginals µ1 and µ2.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Wasserstein distance

Wasserstein distance between measures µ1 and µ2 is dW (µ1, µ2) = sup

f

• fdµ1 −
• fdµ2
• = sup

f

|µ1(f ) − µ2(f )|, where the sup is over measurable 1-Lipschitz functions f : S → R. By the Kantorovich – Rubinstein theorem dW (µ1, µ2) = inf

π {π[d(X, Y )] : L(X) = µ1, L(Y ) = µ2},

with inf over all couplings π on S × S with marginals µ1 and µ2.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

We prove concentration of measure for Lipschitz functions of Xt given bounds on the Wasserstein distance between its i step transition measures Pi for i ≤ t.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Theorem

Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let (αi : i ∈ N) be a sequence of positive constants such that sup

x,y∈S:d(x,y)=1

dW (δxPi, δyPi) ≤ αi, ∀i. (9) Let f be a 1-Lipschitz function. Then for all u > 0, x0 ∈ S, and t > 0, Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

i=0 α2 i ).

(10)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Theorem

Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let (αi : i ∈ N) be a sequence of positive constants such that sup

x,y∈S:d(x,y)=1

dW (δxPi, δyPi) ≤ αi, ∀i. (9) Let f be a 1-Lipschitz function. Then for all u > 0, x0 ∈ S, and t > 0, Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

i=0 α2 i ).

(10)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Theorem

Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let (αi : i ∈ N) be a sequence of positive constants such that sup

x,y∈S:d(x,y)=1

dW (δxPi, δyPi) ≤ αi, ∀i. (9) Let f be a 1-Lipschitz function. Then for all u > 0, x0 ∈ S, and t > 0, Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

i=0 α2 i ).

(10)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Theorem

Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let (αi : i ∈ N) be a sequence of positive constants such that sup

x,y∈S:d(x,y)=1

dW (δxPi, δyPi) ≤ αi, ∀i. (9) Let f be a 1-Lipschitz function. Then for all u > 0, x0 ∈ S, and t > 0, Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

i=0 α2 i ).

(10)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More generally, let S0 be a non-empty subset of S, and let (αi : i ∈ N) be a sequence of positive constants such that, for all i, sup

x,y∈S0:d(x,y)=1

dW (δxPi, δyPi) ≤ αi. (11) Let S0

0 = {x ∈ S0 : y ∈ S0 whenever d(x, y) = 1}.

Let f be a 1-Lipschitz function. Then for all x0 ∈ S0

0, u > 0 and t > 0,

Pδx0

• {|f (Xt) − Eδx0[f (Xt)]| ≥ u} ∩ {Xs ∈ S0

0 : 0 ≤ s ≤ t}

• ≤ 2e−u2/2(Pt−1

i=0 α2 i ). (12) Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More generally, let S0 be a non-empty subset of S, and let (αi : i ∈ N) be a sequence of positive constants such that, for all i, sup

x,y∈S0:d(x,y)=1

dW (δxPi, δyPi) ≤ αi. (11) Let S0

0 = {x ∈ S0 : y ∈ S0 whenever d(x, y) = 1}.

Let f be a 1-Lipschitz function. Then for all x0 ∈ S0

0, u > 0 and t > 0,

Pδx0

• {|f (Xt) − Eδx0[f (Xt)]| ≥ u} ∩ {Xs ∈ S0

0 : 0 ≤ s ≤ t}

• ≤ 2e−u2/2(Pt−1

i=0 α2 i ). (12) Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More generally, let S0 be a non-empty subset of S, and let (αi : i ∈ N) be a sequence of positive constants such that, for all i, sup

x,y∈S0:d(x,y)=1

dW (δxPi, δyPi) ≤ αi. (11) Let S0

0 = {x ∈ S0 : y ∈ S0 whenever d(x, y) = 1}.

Let f be a 1-Lipschitz function. Then for all x0 ∈ S0

0, u > 0 and t > 0,

Pδx0

• {|f (Xt) − Eδx0[f (Xt)]| ≥ u} ∩ {Xs ∈ S0

0 : 0 ≤ s ≤ t}

• ≤ 2e−u2/2(Pt−1

i=0 α2 i ). (12) Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

More generally, let S0 be a non-empty subset of S, and let (αi : i ∈ N) be a sequence of positive constants such that, for all i, sup

x,y∈S0:d(x,y)=1

dW (δxPi, δyPi) ≤ αi. (11) Let S0

0 = {x ∈ S0 : y ∈ S0 whenever d(x, y) = 1}.

Let f be a 1-Lipschitz function. Then for all x0 ∈ S0

0, u > 0 and t > 0,

Pδx0

• {|f (Xt) − Eδx0[f (Xt)]| ≥ u} ∩ {Xs ∈ S0

0 : 0 ≤ s ≤ t}

• ≤ 2e−u2/2(Pt−1

i=0 α2 i ). (12) Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ The following special case satisfying the hypotheses of

Theorem 4 is of particular interest and has received considerable attention in computer science literature;

◮ Suppose (9) holds with αi = αi, for constant 0 < α < 1. ◮ Let (Xt), (X ′ t) be copies with X0 = x and X ′ 0 = x′. ◮ Assume that we can couple (Xt), (X ′ t) so that, uniformly over

all pairs of states x, x′ ∈ S with d(x, x′) = 1, E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ α.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ The following special case satisfying the hypotheses of

Theorem 4 is of particular interest and has received considerable attention in computer science literature;

◮ Suppose (9) holds with αi = αi, for constant 0 < α < 1. ◮ Let (Xt), (X ′ t) be copies with X0 = x and X ′ 0 = x′. ◮ Assume that we can couple (Xt), (X ′ t) so that, uniformly over

all pairs of states x, x′ ∈ S with d(x, x′) = 1, E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ α.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ The following special case satisfying the hypotheses of

Theorem 4 is of particular interest and has received considerable attention in computer science literature;

◮ Suppose (9) holds with αi = αi, for constant 0 < α < 1. ◮ Let (Xt), (X ′ t) be copies with X0 = x and X ′ 0 = x′. ◮ Assume that we can couple (Xt), (X ′ t) so that, uniformly over

all pairs of states x, x′ ∈ S with d(x, x′) = 1, E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ α.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ The following special case satisfying the hypotheses of

Theorem 4 is of particular interest and has received considerable attention in computer science literature;

◮ Suppose (9) holds with αi = αi, for constant 0 < α < 1. ◮ Let (Xt), (X ′ t) be copies with X0 = x and X ′ 0 = x′. ◮ Assume that we can couple (Xt), (X ′ t) so that, uniformly over

all pairs of states x, x′ ∈ S with d(x, x′) = 1, E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ α.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ Thus coupled, (Xt), (X ′ t) will get closer together on average as

t gets larger, which implies strong mixing properties, see Bubley and Dyer (1997), Jerrum (1998).

◮ Then, uniformly over x, x′ ∈ S with d(x, x′) = 1,

dW (δxP, δx′P) ≤ α.

◮ By ‘path coupling’,

E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ αd(x, x′),

i.e. dW (δxP, δx′P) ≤ αd(x, x′) for all pairs x, x′ ∈ S.

◮ By induction on t,

dW (δxPt, δx′Pt) ≤ αtd(x, x′).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ Thus coupled, (Xt), (X ′ t) will get closer together on average as

t gets larger, which implies strong mixing properties, see Bubley and Dyer (1997), Jerrum (1998).

◮ Then, uniformly over x, x′ ∈ S with d(x, x′) = 1,

dW (δxP, δx′P) ≤ α.

◮ By ‘path coupling’,

E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ αd(x, x′),

i.e. dW (δxP, δx′P) ≤ αd(x, x′) for all pairs x, x′ ∈ S.

◮ By induction on t,

dW (δxPt, δx′Pt) ≤ αtd(x, x′).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ Thus coupled, (Xt), (X ′ t) will get closer together on average as

t gets larger, which implies strong mixing properties, see Bubley and Dyer (1997), Jerrum (1998).

◮ Then, uniformly over x, x′ ∈ S with d(x, x′) = 1,

dW (δxP, δx′P) ≤ α.

◮ By ‘path coupling’,

E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ αd(x, x′),

i.e. dW (δxP, δx′P) ≤ αd(x, x′) for all pairs x, x′ ∈ S.

◮ By induction on t,

dW (δxPt, δx′Pt) ≤ αtd(x, x′).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case

◮ Thus coupled, (Xt), (X ′ t) will get closer together on average as

t gets larger, which implies strong mixing properties, see Bubley and Dyer (1997), Jerrum (1998).

◮ Then, uniformly over x, x′ ∈ S with d(x, x′) = 1,

dW (δxP, δx′P) ≤ α.

◮ By ‘path coupling’,

E[d(X1, X ′

1)|X0 = x, X ′ 0 = x′] ≤ αd(x, x′),

i.e. dW (δxP, δx′P) ≤ αd(x, x′) for all pairs x, x′ ∈ S.

◮ By induction on t,

dW (δxPt, δx′Pt) ≤ αtd(x, x′).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case: corollary

Suppose that for 0 < α < 1, dW (δxP, δx′P) ≤ α (13) for all x, x′ ∈ S such that d(x, x′) = 1. Then for all t > 0, u > 0, x0 ∈ S, Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2(1−α2)/2α2 (14) uniformly over 1-Lipschitz functions f on S.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Special case: corollary

Hence, if X has the equilibrium distribution π then, for all u > 0 and every 1-Lipschitz function f , Pπ(|f (X) − Eπ[f (X)]| ≥ u) ≤ 2e−u2(1−α2)/2α2 (15)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Remarks

◮ α = 1 − c1/n corresponds to the ‘optimal’ mixing time

O(n log n) for a Markov chain in a system with size measure n, and gives normal in equilibrium of the form Pπ(|f (Xt) − Eπ[f (Xt)]| ≥ u) ≤ 2e−u2/c2n. (16)

◮ Example: subcritical (β < 1) mean-field Ising model. ◮ Another example: Glauber dynamics for colourings on

bounded-degree graphs - see Dyer et al. (2000,2001,2002) and Molloy (2001).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Remarks

◮ α = 1 − c1/n corresponds to the ‘optimal’ mixing time

O(n log n) for a Markov chain in a system with size measure n, and gives normal in equilibrium of the form Pπ(|f (Xt) − Eπ[f (Xt)]| ≥ u) ≤ 2e−u2/c2n. (16)

◮ Example: subcritical (β < 1) mean-field Ising model. ◮ Another example: Glauber dynamics for colourings on

bounded-degree graphs - see Dyer et al. (2000,2001,2002) and Molloy (2001).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Remarks

◮ α = 1 − c1/n corresponds to the ‘optimal’ mixing time

O(n log n) for a Markov chain in a system with size measure n, and gives normal in equilibrium of the form Pπ(|f (Xt) − Eπ[f (Xt)]| ≥ u) ≤ 2e−u2/c2n. (16)

◮ Example: subcritical (β < 1) mean-field Ising model. ◮ Another example: Glauber dynamics for colourings on

bounded-degree graphs - see Dyer et al. (2000,2001,2002) and Molloy (2001).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Linear extensions of partial order

◮ On the other hand, α = 1 − 6/(n3 − n) for the Glauber

dynamics on linear extensions of a partial order of size n (see Bubley and Dyer) gives an upper bound O(n3 log n) on mixing.

◮ The corresponding bound on deviations of a 1-Lipschitz

function from its mean of size u is of the form 2e−u2/cn3, which is useless, but close to the best possible.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Linear extensions of partial order

◮ On the other hand, α = 1 − 6/(n3 − n) for the Glauber

dynamics on linear extensions of a partial order of size n (see Bubley and Dyer) gives an upper bound O(n3 log n) on mixing.

◮ The corresponding bound on deviations of a 1-Lipschitz

function from its mean of size u is of the form 2e−u2/cn3, which is useless, but close to the best possible.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Consider the partial order on n points consisting of a chain of length n − 1 and a single incomparable element. It is not hard to check that in this case the mixing time is of the order n3. Also there is no meaningful normal concentration of measure bound. (This is similar to a simple random walk on {0, 1, . . . , n}.)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Concentration for non-contracting chains

◮ The choice of α = 1 + c1/n (c1 > 0) is the case where we

cannot couple two copies Xt, X ′

t so they get closer together,

but only so they do not get apart too quickly.

◮ For t = c2n,

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/c3n (17) for all u > 0, all x0 ∈ S, and all 1-Lipschitz functions f .

◮ In some situations, this can be used to establish a law of large

numbers over a fixed time interval for continuous-time Markov chains moving at rate n.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Concentration for non-contracting chains

◮ The choice of α = 1 + c1/n (c1 > 0) is the case where we

cannot couple two copies Xt, X ′

t so they get closer together,

but only so they do not get apart too quickly.

◮ For t = c2n,

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/c3n (17) for all u > 0, all x0 ∈ S, and all 1-Lipschitz functions f .

◮ In some situations, this can be used to establish a law of large

numbers over a fixed time interval for continuous-time Markov chains moving at rate n.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Concentration for non-contracting chains

◮ The choice of α = 1 + c1/n (c1 > 0) is the case where we

cannot couple two copies Xt, X ′

t so they get closer together,

but only so they do not get apart too quickly.

◮ For t = c2n,

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/c3n (17) for all u > 0, all x0 ∈ S, and all 1-Lipschitz functions f .

◮ In some situations, this can be used to establish a law of large

numbers over a fixed time interval for continuous-time Markov chains moving at rate n.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions General theorem Special case: path coupling

Concentration for non-contracting chains

◮ The choice of α = 1 + c1/n (c1 > 0) is the case where we

cannot couple two copies Xt, X ′

t so they get closer together,

but only so they do not get apart too quickly.

◮ For t = c2n,

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/c3n (17) for all u > 0, all x0 ∈ S, and all 1-Lipschitz functions f .

◮ In some situations, this can be used to establish a law of large

numbers over a fixed time interval for continuous-time Markov chains moving at rate n.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ There are n separate queues, each with a single server. ◮ Customers arrive in a Poisson process at rate λn, where

0 < λ < 1 is a constant.

◮ Each new customer chooses d queues uniformly at random

with replacement, and joins a shortest queue amongst those chosen (ties broken by choosing the first of the shortest queues in the list of d).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ There are n separate queues, each with a single server. ◮ Customers arrive in a Poisson process at rate λn, where

0 < λ < 1 is a constant.

◮ Each new customer chooses d queues uniformly at random

with replacement, and joins a shortest queue amongst those chosen (ties broken by choosing the first of the shortest queues in the list of d).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ There are n separate queues, each with a single server. ◮ Customers arrive in a Poisson process at rate λn, where

0 < λ < 1 is a constant.

◮ Each new customer chooses d queues uniformly at random

with replacement, and joins a shortest queue amongst those chosen (ties broken by choosing the first of the shortest queues in the list of d).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ Service times are independent exponentially distributed

random variables with mean 1.

◮ Here d is a fixed positive integer, and we consider large n. ◮ Many authors have studied this model: Mitzenmacher (1996),

Vvedenskaya et al. (1996), Graham (2000,2004), L. et al. (2005,2006,2007)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ Service times are independent exponentially distributed

random variables with mean 1.

◮ Here d is a fixed positive integer, and we consider large n. ◮ Many authors have studied this model: Mitzenmacher (1996),

Vvedenskaya et al. (1996), Graham (2000,2004), L. et al. (2005,2006,2007)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ Service times are independent exponentially distributed

random variables with mean 1.

◮ Here d is a fixed positive integer, and we consider large n. ◮ Many authors have studied this model: Mitzenmacher (1996),

Vvedenskaya et al. (1996), Graham (2000,2004), L. et al. (2005,2006,2007)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ Let X (n) t

• r Xt be the queue-lengths vector

(X (n)

t

(1), . . . , X (n)

t

(n)).

◮ For a positive integer n, (X (n) t

) is an ergodic continuous-time Markov chain, with a unique distribution π(n) or π.

◮ L.and McDiarmid (2006,2007) studied this chain establishing

many results such as rapid mixing (in time of the order log n from good initial states), and concentration of measure for Lipschitz functions long-term and in equilibrium.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ Let X (n) t

• r Xt be the queue-lengths vector

(X (n)

t

(1), . . . , X (n)

t

(n)).

◮ For a positive integer n, (X (n) t

) is an ergodic continuous-time Markov chain, with a unique distribution π(n) or π.

◮ L.and McDiarmid (2006,2007) studied this chain establishing

many results such as rapid mixing (in time of the order log n from good initial states), and concentration of measure for Lipschitz functions long-term and in equilibrium.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model

◮ Let X (n) t

• r Xt be the queue-lengths vector

(X (n)

t

(1), . . . , X (n)

t

(n)).

◮ For a positive integer n, (X (n) t

) is an ergodic continuous-time Markov chain, with a unique distribution π(n) or π.

◮ L.and McDiarmid (2006,2007) studied this chain establishing

many results such as rapid mixing (in time of the order log n from good initial states), and concentration of measure for Lipschitz functions long-term and in equilibrium.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ We now apply our techniques to the jump chain ˆ

X (n)

t

• r ˆ

Xt.

◮ Denote the jump chain stationary measure by ˆ

π(n) or ˆ π.

◮ At time t the next event is an arrival with prob λ/(λ + 1) and

is a potential departure with prob 1/(λ + 1).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ We now apply our techniques to the jump chain ˆ

X (n)

t

• r ˆ

Xt.

◮ Denote the jump chain stationary measure by ˆ

π(n) or ˆ π.

◮ At time t the next event is an arrival with prob λ/(λ + 1) and

is a potential departure with prob 1/(λ + 1).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ We now apply our techniques to the jump chain ˆ

X (n)

t

• r ˆ

Xt.

◮ Denote the jump chain stationary measure by ˆ

π(n) or ˆ π.

◮ At time t the next event is an arrival with prob λ/(λ + 1) and

is a potential departure with prob 1/(λ + 1).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ Given that the next event is an arrival, select a uniformly

random d-tuple of queues and direct the customer to a shortest queue among those chosen.

◮ Given that the next event is a potential departure, choose a

queue uniformly at random. Then a customer will depart if the selected queue is non-empty; otherwise, nothing happens.

◮ It is easy to adapt proofs in L. and McDiarmid (2006) to show

that ˆ Xt is rapidly mixing,

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ Given that the next event is an arrival, select a uniformly

random d-tuple of queues and direct the customer to a shortest queue among those chosen.

◮ Given that the next event is a potential departure, choose a

queue uniformly at random. Then a customer will depart if the selected queue is non-empty; otherwise, nothing happens.

◮ It is easy to adapt proofs in L. and McDiarmid (2006) to show

that ˆ Xt is rapidly mixing,

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ Given that the next event is an arrival, select a uniformly

random d-tuple of queues and direct the customer to a shortest queue among those chosen.

◮ Given that the next event is a potential departure, choose a

queue uniformly at random. Then a customer will depart if the selected queue is non-empty; otherwise, nothing happens.

◮ It is easy to adapt proofs in L. and McDiarmid (2006) to show

that ˆ Xt is rapidly mixing,

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ That is, mixing time is O(n log n) from initial states x with

x 1= O(n) and x ∞= O(log n).

◮ Let ℓ(k, ˆ

Y ) be the number of queues of length at least k in the stationary jump chain, and let ˆ ℓ(k) be its expectation.

◮ Using our inequality, we can show that, for some c > 0,

sup

k∈N

|n−1ℓ(k) − λ1+d+...+dk−1| ≤ cn−1.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ That is, mixing time is O(n log n) from initial states x with

x 1= O(n) and x ∞= O(log n).

◮ Let ℓ(k, ˆ

Y ) be the number of queues of length at least k in the stationary jump chain, and let ˆ ℓ(k) be its expectation.

◮ Using our inequality, we can show that, for some c > 0,

sup

k∈N

|n−1ℓ(k) − λ1+d+...+dk−1| ≤ cn−1.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket model jump chain

◮ That is, mixing time is O(n log n) from initial states x with

x 1= O(n) and x ∞= O(log n).

◮ Let ℓ(k, ˆ

Y ) be the number of queues of length at least k in the stationary jump chain, and let ˆ ℓ(k) be its expectation.

◮ Using our inequality, we can show that, for some c > 0,

sup

k∈N

|n−1ℓ(k) − λ1+d+...+dk−1| ≤ cn−1.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket jump chain concentration

◮ This improves on

sup

k

|n−1ℓ(k) − λ1+d+...+dk−1| ≤ cn−1(log n)2,

• btained by L. and McDiarmid (2006).

◮ Along the way, we improve on the concentration of measure

inequalities for Lipschitz functions established therein.

◮ But the analysis gets quite technical – just as in L. and

McDiarmid (2006), so we shall not discuss the details.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket jump chain concentration

◮ This improves on

sup

k

|n−1ℓ(k) − λ1+d+...+dk−1| ≤ cn−1(log n)2,

• btained by L. and McDiarmid (2006).

◮ Along the way, we improve on the concentration of measure

inequalities for Lipschitz functions established therein.

◮ But the analysis gets quite technical – just as in L. and

McDiarmid (2006), so we shall not discuss the details.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket jump chain concentration

◮ This improves on

sup

k

|n−1ℓ(k) − λ1+d+...+dk−1| ≤ cn−1(log n)2,

• btained by L. and McDiarmid (2006).

◮ Along the way, we improve on the concentration of measure

inequalities for Lipschitz functions established therein.

◮ But the analysis gets quite technical – just as in L. and

McDiarmid (2006), so we shall not discuss the details.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Supermarket: cut-off conjecture

Conjecture

Let c be a positive constant, and let S(n) be the set of all queue lengths vectors x in the n server supermarket model such that x 1≤ cn and x ∞≤ c log n. Let dn(t) = sup

x∈S(n)

dTV(L( ˆ X (n)

t

, x), ˆ π(n)), and let tmix(n) = min{t : dn(t) ≤ 1/4}. Then dn(t) has a cut-off with window size n.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof of Theorem 4

◮ Use a concentration inequality from due to McDiarmid (1998). ◮ Let (˜

Ω, ˜ F, ˜ P) be a finite probability space.

◮ For a σ-field ˜

G ⊆ ˜ F and a bounded random variable Z on (˜ Ω, ˜ F, ˜ P), the supremum of Z in ˜ G is sup(Z| ˜ G)(ω) = min

A∈ ˜ G:ω∈A

max

ω′∈A Z(ω′).

(18)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof of Theorem 4

◮ Use a concentration inequality from due to McDiarmid (1998). ◮ Let (˜

Ω, ˜ F, ˜ P) be a finite probability space.

◮ For a σ-field ˜

G ⊆ ˜ F and a bounded random variable Z on (˜ Ω, ˜ F, ˜ P), the supremum of Z in ˜ G is sup(Z| ˜ G)(ω) = min

A∈ ˜ G:ω∈A

max

ω′∈A Z(ω′).

(18)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof of Theorem 4

◮ Use a concentration inequality from due to McDiarmid (1998). ◮ Let (˜

Ω, ˜ F, ˜ P) be a finite probability space.

◮ For a σ-field ˜

G ⊆ ˜ F and a bounded random variable Z on (˜ Ω, ˜ F, ˜ P), the supremum of Z in ˜ G is sup(Z| ˜ G)(ω) = min

A∈ ˜ G:ω∈A

max

ω′∈A Z(ω′).

(18)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Thus sup(Z | ˜

G) takes the value at ω equal to the maximum value of Z over the ‘smallest’ event in ˜ G containing ω.

◮ The conditional range of Z in ˜

G, denoted by ran(Z), is ran(Z | ˜ G) = sup(Z| ˜ G) + sup(−Z| ˜ G). (19)

◮ Let {∅, ˜

Ω} = ˜ F0 ⊆ ˜ F1 ⊆ . . . ˜ Ft be a filtration in ˜ F.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Thus sup(Z | ˜

G) takes the value at ω equal to the maximum value of Z over the ‘smallest’ event in ˜ G containing ω.

◮ The conditional range of Z in ˜

G, denoted by ran(Z), is ran(Z | ˜ G) = sup(Z| ˜ G) + sup(−Z| ˜ G). (19)

◮ Let {∅, ˜

Ω} = ˜ F0 ⊆ ˜ F1 ⊆ . . . ˜ Ft be a filtration in ˜ F.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Thus sup(Z | ˜

G) takes the value at ω equal to the maximum value of Z over the ‘smallest’ event in ˜ G containing ω.

◮ The conditional range of Z in ˜

G, denoted by ran(Z), is ran(Z | ˜ G) = sup(Z| ˜ G) + sup(−Z| ˜ G). (19)

◮ Let {∅, ˜

Ω} = ˜ F0 ⊆ ˜ F1 ⊆ . . . ˜ Ft be a filtration in ˜ F.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Let Z0, . . . Zt be the martingale Zi = E(Z| ˜

Fi) for each i.

◮ Let rani = ran(Zi| ˜

Fi−1).

◮ Let the sum of squared conditional ranges R2 t be the random

variable t

i=1 ran2 i , and let the maximum sum of squared

conditional ranges ˆ r2

t be the supremum of R2 t :

ˆ r2

t = sup ˜ ω∈˜ Ω

R2

t (˜

ω).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Let Z0, . . . Zt be the martingale Zi = E(Z| ˜

Fi) for each i.

◮ Let rani = ran(Zi| ˜

Fi−1).

◮ Let the sum of squared conditional ranges R2 t be the random

variable t

i=1 ran2 i , and let the maximum sum of squared

conditional ranges ˆ r2

t be the supremum of R2 t :

ˆ r2

t = sup ˜ ω∈˜ Ω

R2

t (˜

ω).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Let Z0, . . . Zt be the martingale Zi = E(Z| ˜

Fi) for each i.

◮ Let rani = ran(Zi| ˜

Fi−1).

◮ Let the sum of squared conditional ranges R2 t be the random

variable t

i=1 ran2 i , and let the maximum sum of squared

conditional ranges ˆ r2

t be the supremum of R2 t :

ˆ r2

t = sup ˜ ω∈˜ Ω

R2

t (˜

ω).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Lemma: concentration of measure inequality

Lemma (McDiarmid (1998))

Let Z be a bounded random variable on a probability space (˜ Ω, ˜ F, ˜ P) with ˜ E(Z) = m. Let {∅, ˜ Ω} = ˜ F0 ⊆ ˜ F1 ⊆ . . . ⊆ ˜ Ft be a filtration in ˜ F, and assume that Z is ˜ Ft-measurable. Then for any u ≥ 0, ˜ P(|Z − m| ≥ u) ≤ 2e−2u2/ˆ

r2

t .

More generally, for any u ≥ 0 and any value r2

t ,

˜ P({|Z − m| ≥ u} ∩ {R2

t ≤ r2 t }) ≤ 2e−2u2/r2

t . Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Let f : S → R be 1-Lipschitz. ◮ Fix a time t ∈ N, x0 ∈ S and consider the evolution of Xt

conditional on X0 = x0 for t steps.

◮ In our setting, we can build the conditional process until time

t on a finite probability space (˜ Ω, ˜ F, ˜ Pδx0): take ˜ Ω to be the finite set of all possible paths of the process starting at time 0 in state x0 until time t, and ˜ F to be the power set of ˜ Ω.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Let f : S → R be 1-Lipschitz. ◮ Fix a time t ∈ N, x0 ∈ S and consider the evolution of Xt

conditional on X0 = x0 for t steps.

◮ In our setting, we can build the conditional process until time

t on a finite probability space (˜ Ω, ˜ F, ˜ Pδx0): take ˜ Ω to be the finite set of all possible paths of the process starting at time 0 in state x0 until time t, and ˜ F to be the power set of ˜ Ω.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ Let f : S → R be 1-Lipschitz. ◮ Fix a time t ∈ N, x0 ∈ S and consider the evolution of Xt

conditional on X0 = x0 for t steps.

◮ In our setting, we can build the conditional process until time

t on a finite probability space (˜ Ω, ˜ F, ˜ Pδx0): take ˜ Ω to be the finite set of all possible paths of the process starting at time 0 in state x0 until time t, and ˜ F to be the power set of ˜ Ω.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ In this space, for each j = 0, . . . , t, let ˜

Fj = σ(X0, . . . , Xj), the σ-field generated by X0, . . . , Xj.

◮ Then ˜

F0 = {∅, ˜ Ω} and ˜ Ft = ˜ F.

◮ Consider Z = f (Xt) : ˜

Ω → R. Also, for j = 0, . . . , t, let Zj be given by Zj = E[f (Xt)| ˜ Fj] = Eδx0[f (Xt)|X0, . . . , Xj] = (Pt−jf )(Xj).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ In this space, for each j = 0, . . . , t, let ˜

Fj = σ(X0, . . . , Xj), the σ-field generated by X0, . . . , Xj.

◮ Then ˜

F0 = {∅, ˜ Ω} and ˜ Ft = ˜ F.

◮ Consider Z = f (Xt) : ˜

Ω → R. Also, for j = 0, . . . , t, let Zj be given by Zj = E[f (Xt)| ˜ Fj] = Eδx0[f (Xt)|X0, . . . , Xj] = (Pt−jf )(Xj).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Proof

◮ In this space, for each j = 0, . . . , t, let ˜

Fj = σ(X0, . . . , Xj), the σ-field generated by X0, . . . , Xj.

◮ Then ˜

F0 = {∅, ˜ Ω} and ˜ Ft = ˜ F.

◮ Consider Z = f (Xt) : ˜

Ω → R. Also, for j = 0, . . . , t, let Zj be given by Zj = E[f (Xt)| ˜ Fj] = Eδx0[f (Xt)|X0, . . . , Xj] = (Pt−jf )(Xj).

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Bounding the ranges

◮ Fix 1 ≤ j ≤ t; we want to upper bound ranj = ran(Zj | ˜

Fj−1).

◮ Fix also x1, . . . , xj−1 ∈ S, and for x ∈ S consider

g(x) = E[f (Xt) | Xj = x] = E[f (Xt−j) | X0 = x] = (Pt−jf )(x).

◮ Zj(˜

ω) ∈ {g(x) : d(x, xj−1) ≤ 1} for ˜ ω s.t. Xj−1(˜ ω) = xj−1.

◮ It follows that, for such ˜

ω, ranj(˜ ω) = sup

x,y:d(x,xj−1)≤1,d(y,xj−1)≤1

|g(x) − g(y)|.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Bounding the ranges

◮ Fix 1 ≤ j ≤ t; we want to upper bound ranj = ran(Zj | ˜

Fj−1).

◮ Fix also x1, . . . , xj−1 ∈ S, and for x ∈ S consider

g(x) = E[f (Xt) | Xj = x] = E[f (Xt−j) | X0 = x] = (Pt−jf )(x).

◮ Zj(˜

ω) ∈ {g(x) : d(x, xj−1) ≤ 1} for ˜ ω s.t. Xj−1(˜ ω) = xj−1.

◮ It follows that, for such ˜

ω, ranj(˜ ω) = sup

x,y:d(x,xj−1)≤1,d(y,xj−1)≤1

|g(x) − g(y)|.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Bounding the ranges

◮ Fix 1 ≤ j ≤ t; we want to upper bound ranj = ran(Zj | ˜

Fj−1).

◮ Fix also x1, . . . , xj−1 ∈ S, and for x ∈ S consider

g(x) = E[f (Xt) | Xj = x] = E[f (Xt−j) | X0 = x] = (Pt−jf )(x).

◮ Zj(˜

ω) ∈ {g(x) : d(x, xj−1) ≤ 1} for ˜ ω s.t. Xj−1(˜ ω) = xj−1.

◮ It follows that, for such ˜

ω, ranj(˜ ω) = sup

x,y:d(x,xj−1)≤1,d(y,xj−1)≤1

|g(x) − g(y)|.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Bounding the ranges

◮ Fix 1 ≤ j ≤ t; we want to upper bound ranj = ran(Zj | ˜

Fj−1).

◮ Fix also x1, . . . , xj−1 ∈ S, and for x ∈ S consider

g(x) = E[f (Xt) | Xj = x] = E[f (Xt−j) | X0 = x] = (Pt−jf )(x).

◮ Zj(˜

ω) ∈ {g(x) : d(x, xj−1) ≤ 1} for ˜ ω s.t. Xj−1(˜ ω) = xj−1.

◮ It follows that, for such ˜

ω, ranj(˜ ω) = sup

x,y:d(x,xj−1)≤1,d(y,xj−1)≤1

|g(x) − g(y)|.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Bounding the ranges

As f is 1-Lipschitz, supx,y:d(x,y)≤2 |g(x) − g(y)| is at most sup

x,y:d(x,y)≤2

|(Pt−jf )(x) − (Pt−jf )(y)| = sup

x,y:d(x,y)≤2

| EδxPt−j(f ) − EδyPt−j(f )| ≤ 2 sup

x,y:d(x,y)≤1

| EδxPt−j(f ) − EδyPt−j(f )| ≤ 2 sup

x,y:d(x,y)≤1

dW (δxPt−j, δyPt−j) ≤ 2αt−j.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

◮ Hence ranj(˜

ω) ≤ 2αt−j for all ˜ ω ∈ ˜ Ω.

◮ And hence, uniformly over ˜

ω ∈ ˜ Ω, ˆ r2

t (˜

ω) ≤ 4

t

• j=1

α2

t−j = t−1

• j=0

α2

j . ◮ Now apply Lemma 6 to deduce part (i) of the theorem:

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

j=0 α2 j ). Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

◮ Hence ranj(˜

ω) ≤ 2αt−j for all ˜ ω ∈ ˜ Ω.

◮ And hence, uniformly over ˜

ω ∈ ˜ Ω, ˆ r2

t (˜

ω) ≤ 4

t

• j=1

α2

t−j = t−1

• j=0

α2

j . ◮ Now apply Lemma 6 to deduce part (i) of the theorem:

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

j=0 α2 j ). Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

◮ Hence ranj(˜

ω) ≤ 2αt−j for all ˜ ω ∈ ˜ Ω.

◮ And hence, uniformly over ˜

ω ∈ ˜ Ω, ˆ r2

t (˜

ω) ≤ 4

t

• j=1

α2

t−j = t−1

• j=0

α2

j . ◮ Now apply Lemma 6 to deduce part (i) of the theorem:

Pδx0(|f (Xt) − Eδx0[f (Xt)]| ≥ u) ≤ 2e−u2/2(Pt−1

j=0 α2 j ). Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

End of proof

To prove part (ii), observe that the bound ranj(ω) = ran(Zj | ˜ Fj−1)(ω) ≤ 2αt−j still holds on the event At = {ω : Xj(ω) ∈ S0

0 for j = 0, . . . , t}.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Conclusions

◮ We have derived concentration inequalities for Lipschitz

functions of a Markov chain long-term and in equilibrium, depending on the contractivity of the chain.

◮ Our results apply to many natural Markov chains in computer

science and statistical mechanics.

◮ Open problem: show that in an MC with ‘local’ transitions,

under suitable conditions, rapid mixing occurs essentially if and only if there is normal concentration of measure long-term and in equilibrium (with non-trivial bounds).

◮ Open problem: under suitable assumptions, are these

necessary and sufficient conditions for a cut-off to occur?

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Conclusions

◮ We have derived concentration inequalities for Lipschitz

functions of a Markov chain long-term and in equilibrium, depending on the contractivity of the chain.

◮ Our results apply to many natural Markov chains in computer

science and statistical mechanics.

◮ Open problem: show that in an MC with ‘local’ transitions,

under suitable conditions, rapid mixing occurs essentially if and only if there is normal concentration of measure long-term and in equilibrium (with non-trivial bounds).

◮ Open problem: under suitable assumptions, are these

necessary and sufficient conditions for a cut-off to occur?

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Conclusions

◮ We have derived concentration inequalities for Lipschitz

functions of a Markov chain long-term and in equilibrium, depending on the contractivity of the chain.

◮ Our results apply to many natural Markov chains in computer

science and statistical mechanics.

◮ Open problem: show that in an MC with ‘local’ transitions,

under suitable conditions, rapid mixing occurs essentially if and only if there is normal concentration of measure long-term and in equilibrium (with non-trivial bounds).

◮ Open problem: under suitable assumptions, are these

necessary and sufficient conditions for a cut-off to occur?

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Conclusions

◮ We have derived concentration inequalities for Lipschitz

functions of a Markov chain long-term and in equilibrium, depending on the contractivity of the chain.

◮ Our results apply to many natural Markov chains in computer

science and statistical mechanics.

◮ Open problem: show that in an MC with ‘local’ transitions,

under suitable conditions, rapid mixing occurs essentially if and only if there is normal concentration of measure long-term and in equilibrium (with non-trivial bounds).

◮ Open problem: under suitable assumptions, are these

necessary and sufficient conditions for a cut-off to occur?

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollaries

Corollary

Suppose that there exists x ∈ S and a sequence αi : S → R+ of functions such that, for all y ∈ S, dW (δxPi, δyPi) ≤ αi(y), where αi(y) → 0 as i → ∞ for each y, and supk Eδx[αi(Xk)] = supk(Pkαi)(x) → 0 as i → ∞. Then (Xt) has a unique stationary measure π and, for all y ∈ S, δyPt → π as t → ∞.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollaries

Corollary

Suppose that there exists x ∈ S and a sequence αi : S → R+ of functions such that, for all y ∈ S, dW (δxPi, δyPi) ≤ αi(y), where αi(y) → 0 as i → ∞ for each y, and supk Eδx[αi(Xk)] = supk(Pkαi)(x) → 0 as i → ∞. Then (Xt) has a unique stationary measure π and, for all y ∈ S, δyPt → π as t → ∞.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollaries

Corollary

Suppose that there exists x ∈ S and a sequence αi : S → R+ of functions such that, for all y ∈ S, dW (δxPi, δyPi) ≤ αi(y), where αi(y) → 0 as i → ∞ for each y, and supk Eδx[αi(Xk)] = supk(Pkαi)(x) → 0 as i → ∞. Then (Xt) has a unique stationary measure π and, for all y ∈ S, δyPt → π as t → ∞.

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose (9) holds, and the αi in Theorem 4 satisfy

i α2 i < ∞.

Let g(y) = d(x, y), and suppose that for some x ∈ S supk(Pkg)(x) < ∞. Then (Xt) has a unique stationary measure π, δyPt → π as t → ∞ for each y. Furthermore, let X be a stationary copy of Xt. Then, for all u > 0, and uniformly over all 1-Lipschitz functions f , Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(P∞

i=1 α2 i ).

(20)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose (9) holds, and the αi in Theorem 4 satisfy

i α2 i < ∞.

Let g(y) = d(x, y), and suppose that for some x ∈ S supk(Pkg)(x) < ∞. Then (Xt) has a unique stationary measure π, δyPt → π as t → ∞ for each y. Furthermore, let X be a stationary copy of Xt. Then, for all u > 0, and uniformly over all 1-Lipschitz functions f , Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(P∞

i=1 α2 i ).

(20)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose (9) holds, and the αi in Theorem 4 satisfy

i α2 i < ∞.

Let g(y) = d(x, y), and suppose that for some x ∈ S supk(Pkg)(x) < ∞. Then (Xt) has a unique stationary measure π, δyPt → π as t → ∞ for each y. Furthermore, let X be a stationary copy of Xt. Then, for all u > 0, and uniformly over all 1-Lipschitz functions f , Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(P∞

i=1 α2 i ).

(20)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose (9) holds, and the αi in Theorem 4 satisfy

i α2 i < ∞.

Let g(y) = d(x, y), and suppose that for some x ∈ S supk(Pkg)(x) < ∞. Then (Xt) has a unique stationary measure π, δyPt → π as t → ∞ for each y. Furthermore, let X be a stationary copy of Xt. Then, for all u > 0, and uniformly over all 1-Lipschitz functions f , Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(P∞

i=1 α2 i ).

(20)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose that (Xt) has a unique stationary measure π and condition (11) holds, where

i α2 i < ∞.

Let x ∈ S0

0, and suppose δ > 0 and t0 > 0 are such that

dW (δxPt0, π) < δ. Also suppose Pδx(Xt ∈ S0

0 for t ≤ t0) ≥ 1 − δ.

Let X be a stationary copy of Xt. Then, for all u ≥ δ, uniformly

• ver all 1-Lipschitz functions f ,

Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(Pt0

i=1 α2 i ) + 2δ.

(21)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose that (Xt) has a unique stationary measure π and condition (11) holds, where

i α2 i < ∞.

Let x ∈ S0

0, and suppose δ > 0 and t0 > 0 are such that

dW (δxPt0, π) < δ. Also suppose Pδx(Xt ∈ S0

0 for t ≤ t0) ≥ 1 − δ.

Let X be a stationary copy of Xt. Then, for all u ≥ δ, uniformly

• ver all 1-Lipschitz functions f ,

Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(Pt0

i=1 α2 i ) + 2δ.

(21)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose that (Xt) has a unique stationary measure π and condition (11) holds, where

i α2 i < ∞.

Let x ∈ S0

0, and suppose δ > 0 and t0 > 0 are such that

dW (δxPt0, π) < δ. Also suppose Pδx(Xt ∈ S0

0 for t ≤ t0) ≥ 1 − δ.

Let X be a stationary copy of Xt. Then, for all u ≥ δ, uniformly

• ver all 1-Lipschitz functions f ,

Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(Pt0

i=1 α2 i ) + 2δ.

(21)

Malwina J Luczak Concentration of measure and mixing for Markov chains

Introduction Mean-field Ising model Generalisation Supermarket model Proof Conclusions

Corollary

Suppose that (Xt) has a unique stationary measure π and condition (11) holds, where

i α2 i < ∞.

Let x ∈ S0

0, and suppose δ > 0 and t0 > 0 are such that

dW (δxPt0, π) < δ. Also suppose Pδx(Xt ∈ S0

0 for t ≤ t0) ≥ 1 − δ.

Let X be a stationary copy of Xt. Then, for all u ≥ δ, uniformly

• ver all 1-Lipschitz functions f ,

Pπ(|f (X) − Eπ[f (X)]| ≥ 2u) ≤ 2e−u2/2(Pt0

i=1 α2 i ) + 2δ.

(21)

Malwina J Luczak Concentration of measure and mixing for Markov chains