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 σ ( v ) σ ( w ) , (1) v , w ∈ V , v ∼ w ◮ 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 σ ( v ) σ ( w ) , (1) v , w ∈ V , v ∼ w ◮ 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 σ ( v ) σ ( w ) , (1) v , w ∈ V , v ∼ w ◮ 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 σ ( v ) σ ( w ) , (1) v , w ∈ V , v ∼ w ◮ 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 − β H ( σ ) Z ( β ) . (2) σ ∈ S e − β H ( σ ) is a normalising constant. ◮ Here Z ( β ) = � ◮ 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 over 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 − β H ( σ ) Z ( β ) . (2) σ ∈ S e − β H ( σ ) is a normalising constant. ◮ Here Z ( β ) = � ◮ 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 over 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 − β H ( σ ) Z ( β ) . (2) σ ∈ S e − β H ( σ ) is a normalising constant. ◮ Here Z ( β ) = � ◮ 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 over 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 − β H ( σ ) Z ( β ) . (2) σ ∈ S e − β H ( σ ) is a normalising constant. ◮ Here Z ( β ) = � ◮ 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 over 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 ( X t ) 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 e β M v ( σ ) p ( σ ; v ) := e β M v ( σ ) + e − β M v ( σ ) , (3) with M v ( 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 ( X t ) 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 e β M v ( σ ) p ( σ ; v ) := e β M v ( σ ) + e − β M v ( σ ) , (3) with M v ( 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 ( X t ) 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 e β M v ( σ ) p ( σ ; v ) := e β M v ( σ ) + e − β M v ( σ ) , (3) with M v ( 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 on the current spins at the neighbours of v . ◮ ( X t ) is reversible with respect to Gibbs measure π , which is thus its stationary measure. ◮ Given a sequence G n = ( V n , E n ) of graphs, let π n be the Ising measure and ( X ( n ) ) the Glauber dynamics on G n . 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 ◮ Evidently, the distribution of the new spin at v depends only on the current spins at the neighbours of v . ◮ ( X t ) is reversible with respect to Gibbs measure π , which is thus its stationary measure. ◮ Given a sequence G n = ( V n , E n ) of graphs, let π n be the Ising measure and ( X ( n ) ) the Glauber dynamics on G n . 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 ◮ Evidently, the distribution of the new spin at v depends only on the current spins at the neighbours of v . ◮ ( X t ) is reversible with respect to Gibbs measure π , which is thus its stationary measure. ◮ Given a sequence G n = ( V n , E n ) of graphs, let π n be the Ising measure and ( X ( n ) ) the Glauber dynamics on G n . 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 ◮ For σ ∈ S n , let L ( X ( n ) , σ ) be the law of X ( n ) starting from σ . t t ◮ The worst-case distance to stationarity of the Glauber dynamics chain after t steps is σ ∈ S n d TV ( L ( X ( n ) d n ( t ) := max , σ ) , π n ) . (4) 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 ◮ For σ ∈ S n , let L ( X ( n ) , σ ) be the law of X ( n ) starting from σ . t t ◮ The worst-case distance to stationarity of the Glauber dynamics chain after t steps is σ ∈ S n d TV ( L ( X ( n ) d n ( t ) := max , σ ) , π n ) . (4) 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 t mix ( n ) is defined as t mix ( n ) := min { t : d n ( t ) ≤ 1 / 4 } . (5) ◮ Note t mix ( n ) is finite for each fixed n since, by the ergodic theorem for Markov chains, d n ( 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 t mix ( n ) is defined as t mix ( n ) := min { t : d n ( t ) ≤ 1 / 4 } . (5) ◮ Note t mix ( n ) is finite for each fixed n since, by the ergodic theorem for Markov chains, d n ( 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 t mix ( n ) will in general tend to infinity with n , and it is natural to ask about the growth rate of t mix ( n ). Definition The Glauber dynamics is said to exhibit a cut-off at { t n } with window size { w n } if w n = o ( t n ) and γ →∞ lim inf lim n →∞ d n ( t n − γ w n ) = 1 , γ →∞ lim sup lim n →∞ d n ( t n + γ w n ) = 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 t mix ( n ) will in general tend to infinity with n , and it is natural to ask about the growth rate of t mix ( n ). Definition The Glauber dynamics is said to exhibit a cut-off at { t n } with window size { w n } if w n = o ( t n ) and γ →∞ lim inf lim n →∞ d n ( t n − γ w n ) = 1 , γ →∞ lim sup lim n →∞ d n ( t n + γ w n ) = 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 t mix ( n ) will in general tend to infinity with n , and it is natural to ask about the growth rate of t mix ( n ). Definition The Glauber dynamics is said to exhibit a cut-off at { t n } with window size { w n } if w n = o ( t n ) and γ →∞ lim inf lim n →∞ d n ( t n − γ w n ) = 1 , γ →∞ lim sup lim n →∞ d n ( t n + γ w n ) = 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. G n is K n , the complete graph on n vertices. ◮ Take J = 1 / n ; so measure π on {− 1 , 1 } n is 1 β � . π ( σ ) = π n ( σ ) = Z ( β ) exp σ ( i ) σ ( j ) (6) n 1 ≤ i < j ≤ n ◮ We update to a +1 with prob. p + ( S ( σ ) − n − 1 σ ( i )), where e β s p + ( s ) = e β s + e − β s . 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. G n is K n , the complete graph on n vertices. ◮ Take J = 1 / n ; so measure π on {− 1 , 1 } n is 1 β � . π ( σ ) = π n ( σ ) = Z ( β ) exp σ ( i ) σ ( j ) (6) n 1 ≤ i < j ≤ n ◮ We update to a +1 with prob. p + ( S ( σ ) − n − 1 σ ( i )), where e β s p + ( s ) = e β s + e − β s . 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. G n is K n , the complete graph on n vertices. ◮ Take J = 1 / n ; so measure π on {− 1 , 1 } n is 1 β � . π ( σ ) = π n ( σ ) = Z ( β ) exp σ ( i ) σ ( j ) (6) n 1 ≤ i < j ≤ n ◮ We update to a +1 with prob. p + ( S ( σ ) − n − 1 σ ( i )), where e β s p + ( s ) = e β s + e − β s . 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 t mix ( 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 ℓ 1 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 t mix ( 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 ℓ 1 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 t mix ( 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 ℓ 1 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 ) in equilibrium. t ◮ Then, for some c , C > 0, P π ( | f ( X ( n ) ) − E π f ( X ( n ) ) | ≥ u ) ≤ Ce − u 2 / 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 − u 2 / 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 ) in equilibrium. t ◮ Then, for some c , C > 0, P π ( | f ( X ( n ) ) − E π f ( X ( n ) ) | ≥ u ) ≤ Ce − u 2 / 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 − u 2 / 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 ) in equilibrium. t ◮ Then, for some c , C > 0, P π ( | f ( X ( n ) ) − E π f ( X ( n ) ) | ≥ u ) ≤ Ce − u 2 / 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 − u 2 / 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 K n has a cut-off at t n = [2(1 − β )] − 1 n log n with window size n. ◮ Furthermore, asymptotically the spin values in a bounded set of 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 K n has a cut-off at t n = [2(1 − β )] − 1 n log n with window size n. ◮ Furthermore, asymptotically the spin values in a bounded set of 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 K n has a cut-off at t n = [2(1 − β )] − 1 n log n with window size n. ◮ Furthermore, asymptotically the spin values in a bounded set of 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 t mix ( n ) is polynomial in n . Theorem [Levin, L. and Peres (2008)] Let β = 1 . There are C 1 , C 2 > 0 such that C 1 n 3 / 2 ≤ t mix ( n ) ≤ C 2 n 3 / 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 t mix ( n ) is polynomial in n . Theorem [Levin, L. and Peres (2008)] Let β = 1 . There are C 1 , C 2 > 0 such that C 1 n 3 / 2 ≤ t mix ( n ) ≤ C 2 n 3 / 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 of 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 of 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 of 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 t mix ( n ) is exponential in n . This can be proved e.g. by upper bounding the Cheeger constant � σ ∈ A ,τ �∈ A π ( σ ) P ( σ, τ ) Φ = min . π ( A ) A : π ( A ) ≤ 1 / 2 ◮ 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 t mix ( n ) is exponential in n . This can be proved e.g. by upper bounding the Cheeger constant � σ ∈ A ,τ �∈ A π ( σ ) P ( σ, τ ) Φ = min . π ( A ) A : π ( A ) ≤ 1 / 2 ◮ 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 t mix ( n ) is exponential in n . This can be proved e.g. by upper bounding the Cheeger constant � σ ∈ A ,τ �∈ A π ( σ ) P ( σ, τ ) Φ = min . π ( A ) A : π ( A ) ≤ 1 / 2 ◮ 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 General theorem Supermarket model Special case: path coupling Proof Conclusions Our general setting ◮ X = ( X t ) 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 General theorem Supermarket model Special case: path coupling Proof Conclusions Our general setting ◮ X = ( X t ) 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 General theorem Supermarket model Special case: path coupling Proof Conclusions Our general setting ◮ X = ( X t ) 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 General theorem Supermarket model Special case: path coupling Proof Conclusions Our general setting ◮ X = ( X t ) 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 General theorem Supermarket model Special case: path coupling Proof Conclusions General setting continued ◮ X t 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 General theorem Supermarket model Special case: path coupling Proof Conclusions General setting continued ◮ X t 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 General theorem Supermarket model Special case: path coupling Proof Conclusions General setting continued ◮ X t 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 General theorem Supermarket model Special case: path coupling Proof Conclusions General setting continued ◮ X t 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 General theorem Supermarket model Special case: path coupling Proof Conclusions Underlying probability space ◮ Each X t can be seen as a r.v. on (Ω , F ), where Ω = { ω = ( ω 0 , ω 1 , . . . ) : ω i ∈ S ∀ i } , and X t is the t -co-ordinate projection, i.e. X t ( ω ) = ω t . ◮ Also, F = σ ( ∪ ∞ t =0 F t ), with F t = σ ( X i : i ≤ t ). ◮ Law of ( X t ) can be viewed as a probability measure P on (Ω , F ): t − 1 � P ( { ω : ω j = x j : j ≤ t } ) = µ ( { x 0 } ) P ( x j , x j +1 ) , j =0 Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Underlying probability space ◮ Each X t can be seen as a r.v. on (Ω , F ), where Ω = { ω = ( ω 0 , ω 1 , . . . ) : ω i ∈ S ∀ i } , and X t is the t -co-ordinate projection, i.e. X t ( ω ) = ω t . ◮ Also, F = σ ( ∪ ∞ t =0 F t ), with F t = σ ( X i : i ≤ t ). ◮ Law of ( X t ) can be viewed as a probability measure P on (Ω , F ): t − 1 � P ( { ω : ω j = x j : j ≤ t } ) = µ ( { x 0 } ) P ( x j , x j +1 ) , j =0 Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Underlying probability space ◮ Each X t can be seen as a r.v. on (Ω , F ), where Ω = { ω = ( ω 0 , ω 1 , . . . ) : ω i ∈ S ∀ i } , and X t is the t -co-ordinate projection, i.e. X t ( ω ) = ω t . ◮ Also, F = σ ( ∪ ∞ t =0 F t ), with F t = σ ( X i : i ≤ t ). ◮ Law of ( X t ) can be viewed as a probability measure P on (Ω , F ): t − 1 � P ( { ω : ω j = x j : j ≤ t } ) = µ ( { x 0 } ) P ( x j , x j +1 ) , j =0 Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More definitions ◮ This is P µ , law of ( X t ) conditional on L ( X 0 ) = µ . ◮ P t ( x , y ) is the t -step transition probability from x to y , given inductively by � P t − 1 ( x , z ) P ( z , y ) . P t ( x , y ) = z ∈ S ◮ Then P µ ( X t ∈ A ) = ( µ P t )( A ) for A ⊆ S . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More definitions ◮ This is P µ , law of ( X t ) conditional on L ( X 0 ) = µ . ◮ P t ( x , y ) is the t -step transition probability from x to y , given inductively by � P t − 1 ( x , z ) P ( z , y ) . P t ( x , y ) = z ∈ S ◮ Then P µ ( X t ∈ A ) = ( µ P t )( A ) for A ⊆ S . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More definitions ◮ This is P µ , law of ( X t ) conditional on L ( X 0 ) = µ . ◮ P t ( x , y ) is the t -step transition probability from x to y , given inductively by � P t − 1 ( x , z ) P ( z , y ) . P t ( x , y ) = z ∈ S ◮ Then P µ ( X t ∈ A ) = ( µ P t )( A ) for A ⊆ S . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More definitions ◮ E µ is the corresponding expectation operator. ◮ For t ∈ Z + and f : S → R , � ( P t f )( x ) = P t ( x , y ) f ( y ) , x ∈ S . y ◮ So ( P t f )( x ) = E δ x [ f ( X t )] = ( δ x P t )( f ), the expected value of f ( X t ) conditional on X t starting at x . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More definitions ◮ E µ is the corresponding expectation operator. ◮ For t ∈ Z + and f : S → R , � ( P t f )( x ) = P t ( x , y ) f ( y ) , x ∈ S . y ◮ So ( P t f )( x ) = E δ x [ f ( X t )] = ( δ x P t )( f ), the expected value of f ( X t ) conditional on X t starting at x . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More definitions ◮ E µ is the corresponding expectation operator. ◮ For t ∈ Z + and f : S → R , � ( P t f )( x ) = P t ( x , y ) f ( y ) , x ∈ S . y ◮ So ( P t f )( x ) = E δ x [ f ( X t )] = ( δ x P t )( f ), the expected value of f ( X t ) conditional on X t starting at x . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Lipschitz functions ◮ Function f : S → R is Lipschitz (or 1-Lipschtitz) if | f ( x ) − f ( y ) | � f � Lip = sup ≤ 1 . d ( x , y ) x � = y ◮ Equivalently, f is Lipschitz if sup x , 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 General theorem Supermarket model Special case: path coupling Proof Conclusions Lipschitz functions ◮ Function f : S → R is Lipschitz (or 1-Lipschtitz) if | f ( x ) − f ( y ) | � f � Lip = sup ≤ 1 . d ( x , y ) x � = y ◮ Equivalently, f is Lipschitz if sup x , 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 General theorem Supermarket model Special case: path coupling Proof Conclusions 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 − cu 2 . (8) ◮ We shall give conditions when ( X t ) has normal concentration of measure long term and in equilibrium. Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions 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 − cu 2 . (8) ◮ We shall give conditions when ( X t ) has normal concentration of measure long term and in equilibrium. Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Wasserstein distance Wasserstein distance between measures µ 1 and µ 2 is � � � � � � fd µ 1 − | µ 1 ( f ) − µ 2 ( f ) | , d W ( µ 1 , µ 2 ) = sup fd µ 2 � = sup � � � f f where the sup is over measurable 1-Lipschitz functions f : S → R . By the Kantorovich – Rubinstein theorem π { π [ d ( X , Y )] : L ( X ) = µ 1 , L ( Y ) = µ 2 } , d W ( µ 1 , µ 2 ) = inf with inf over all couplings π on S × S with marginals µ 1 and µ 2 . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Wasserstein distance Wasserstein distance between measures µ 1 and µ 2 is � � � � � � fd µ 1 − | µ 1 ( f ) − µ 2 ( f ) | , d W ( µ 1 , µ 2 ) = sup fd µ 2 � = sup � � � f f where the sup is over measurable 1-Lipschitz functions f : S → R . By the Kantorovich – Rubinstein theorem π { π [ d ( X , Y )] : L ( X ) = µ 1 , L ( Y ) = µ 2 } , d W ( µ 1 , µ 2 ) = inf with inf over all couplings π on S × S with marginals µ 1 and µ 2 . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions We prove concentration of measure for Lipschitz functions of X t given bounds on the Wasserstein distance between its i step transition measures P i for i ≤ t . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Theorem Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let ( α i : i ∈ N ) be a sequence of positive constants such that d W ( δ x P i , δ y P i ) ≤ α i , ∀ i . sup (9) x , y ∈ S : d ( x , y )=1 Let f be a 1-Lipschitz function. Then for all u > 0 , x 0 ∈ S, and t > 0 , P δ x 0 ( | f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u ) ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (10) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Theorem Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let ( α i : i ∈ N ) be a sequence of positive constants such that d W ( δ x P i , δ y P i ) ≤ α i , ∀ i . sup (9) x , y ∈ S : d ( x , y )=1 Let f be a 1-Lipschitz function. Then for all u > 0 , x 0 ∈ S, and t > 0 , P δ x 0 ( | f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u ) ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (10) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Theorem Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let ( α i : i ∈ N ) be a sequence of positive constants such that d W ( δ x P i , δ y P i ) ≤ α i , ∀ i . sup (9) x , y ∈ S : d ( x , y )=1 Let f be a 1-Lipschitz function. Then for all u > 0 , x 0 ∈ S, and t > 0 , P δ x 0 ( | f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u ) ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (10) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Theorem Let P be the transition matrix of a discrete-time Markov chain with discrete state space S. Let ( α i : i ∈ N ) be a sequence of positive constants such that d W ( δ x P i , δ y P i ) ≤ α i , ∀ i . sup (9) x , y ∈ S : d ( x , y )=1 Let f be a 1-Lipschitz function. Then for all u > 0 , x 0 ∈ S, and t > 0 , P δ x 0 ( | f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u ) ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (10) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More generally, let S 0 be a non-empty subset of S , and let ( α i : i ∈ N ) be a sequence of positive constants such that, for all i , d W ( δ x P i , δ y P i ) ≤ α i . sup (11) x , y ∈ S 0 : d ( x , y )=1 Let S 0 0 = { x ∈ S 0 : y ∈ S 0 whenever d ( x , y ) = 1 } . Let f be a 1-Lipschitz function. Then for all x 0 ∈ S 0 0 , u > 0 and t > 0, � � {| f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u } ∩ { X s ∈ S 0 0 : 0 ≤ s ≤ t } P δ x 0 ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (12) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More generally, let S 0 be a non-empty subset of S , and let ( α i : i ∈ N ) be a sequence of positive constants such that, for all i , d W ( δ x P i , δ y P i ) ≤ α i . sup (11) x , y ∈ S 0 : d ( x , y )=1 Let S 0 0 = { x ∈ S 0 : y ∈ S 0 whenever d ( x , y ) = 1 } . Let f be a 1-Lipschitz function. Then for all x 0 ∈ S 0 0 , u > 0 and t > 0, � � {| f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u } ∩ { X s ∈ S 0 0 : 0 ≤ s ≤ t } P δ x 0 ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (12) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More generally, let S 0 be a non-empty subset of S , and let ( α i : i ∈ N ) be a sequence of positive constants such that, for all i , d W ( δ x P i , δ y P i ) ≤ α i . sup (11) x , y ∈ S 0 : d ( x , y )=1 Let S 0 0 = { x ∈ S 0 : y ∈ S 0 whenever d ( x , y ) = 1 } . Let f be a 1-Lipschitz function. Then for all x 0 ∈ S 0 0 , u > 0 and t > 0, � � {| f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u } ∩ { X s ∈ S 0 0 : 0 ≤ s ≤ t } P δ x 0 ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (12) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions More generally, let S 0 be a non-empty subset of S , and let ( α i : i ∈ N ) be a sequence of positive constants such that, for all i , d W ( δ x P i , δ y P i ) ≤ α i . sup (11) x , y ∈ S 0 : d ( x , y )=1 Let S 0 0 = { x ∈ S 0 : y ∈ S 0 whenever d ( x , y ) = 1 } . Let f be a 1-Lipschitz function. Then for all x 0 ∈ S 0 0 , u > 0 and t > 0, � � {| f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u } ∩ { X s ∈ S 0 0 : 0 ≤ s ≤ t } P δ x 0 ≤ 2 e − u 2 / 2( P t − 1 i =0 α 2 i ) . (12) Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions 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 α i = α i , for constant 0 < α < 1. ◮ Let ( X t ) , ( X ′ t ) be copies with X 0 = x and X ′ 0 = x ′ . ◮ Assume that we can couple ( X t ) , ( X ′ t ) so that, uniformly over all pairs of states x , x ′ ∈ S with d ( x , x ′ ) = 1, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α. Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions 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 α i = α i , for constant 0 < α < 1. ◮ Let ( X t ) , ( X ′ t ) be copies with X 0 = x and X ′ 0 = x ′ . ◮ Assume that we can couple ( X t ) , ( X ′ t ) so that, uniformly over all pairs of states x , x ′ ∈ S with d ( x , x ′ ) = 1, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α. Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions 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 α i = α i , for constant 0 < α < 1. ◮ Let ( X t ) , ( X ′ t ) be copies with X 0 = x and X ′ 0 = x ′ . ◮ Assume that we can couple ( X t ) , ( X ′ t ) so that, uniformly over all pairs of states x , x ′ ∈ S with d ( x , x ′ ) = 1, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α. Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions 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 α i = α i , for constant 0 < α < 1. ◮ Let ( X t ) , ( X ′ t ) be copies with X 0 = x and X ′ 0 = x ′ . ◮ Assume that we can couple ( X t ) , ( X ′ t ) so that, uniformly over all pairs of states x , x ′ ∈ S with d ( x , x ′ ) = 1, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α. Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Special case ◮ Thus coupled, ( X t ) , ( 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, d W ( δ x P , δ x ′ P ) ≤ α . ◮ By ‘path coupling’, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α d ( x , x ′ ) , i.e. d W ( δ x P , δ x ′ P ) ≤ α d ( x , x ′ ) for all pairs x , x ′ ∈ S . ◮ By induction on t , d W ( δ x P t , δ x ′ P t ) ≤ α t d ( x , x ′ ) . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Special case ◮ Thus coupled, ( X t ) , ( 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, d W ( δ x P , δ x ′ P ) ≤ α . ◮ By ‘path coupling’, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α d ( x , x ′ ) , i.e. d W ( δ x P , δ x ′ P ) ≤ α d ( x , x ′ ) for all pairs x , x ′ ∈ S . ◮ By induction on t , d W ( δ x P t , δ x ′ P t ) ≤ α t d ( x , x ′ ) . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Special case ◮ Thus coupled, ( X t ) , ( 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, d W ( δ x P , δ x ′ P ) ≤ α . ◮ By ‘path coupling’, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α d ( x , x ′ ) , i.e. d W ( δ x P , δ x ′ P ) ≤ α d ( x , x ′ ) for all pairs x , x ′ ∈ S . ◮ By induction on t , d W ( δ x P t , δ x ′ P t ) ≤ α t d ( x , x ′ ) . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Special case ◮ Thus coupled, ( X t ) , ( 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, d W ( δ x P , δ x ′ P ) ≤ α . ◮ By ‘path coupling’, E [ d ( X 1 , X ′ 1 ) | X 0 = x , X ′ 0 = x ′ ] ≤ α d ( x , x ′ ) , i.e. d W ( δ x P , δ x ′ P ) ≤ α d ( x , x ′ ) for all pairs x , x ′ ∈ S . ◮ By induction on t , d W ( δ x P t , δ x ′ P t ) ≤ α t d ( x , x ′ ) . Malwina J Luczak Concentration of measure and mixing for Markov chains
Introduction Mean-field Ising model Generalisation General theorem Supermarket model Special case: path coupling Proof Conclusions Special case: corollary Suppose that for 0 < α < 1, d W ( δ x P , δ x ′ P ) ≤ α (13) for all x , x ′ ∈ S such that d ( x , x ′ ) = 1. Then for all t > 0, u > 0, x 0 ∈ S , P δ x 0 ( | f ( X t ) − E δ x 0 [ f ( X t )] | ≥ u ) ≤ 2 e − u 2 (1 − α 2 ) / 2 α 2 (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 General theorem Supermarket model Special case: path coupling Proof Conclusions Special case: corollary Hence, if X has the equilibrium distribution π then, for all u > 0 and every 1-Lipschitz function f , P π ( | f ( X ) − E π [ f ( X )] | ≥ u ) ≤ 2 e − u 2 (1 − α 2 ) / 2 α 2 (15) Malwina J Luczak Concentration of measure and mixing for Markov chains
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