Convex decay of Entropy for interacting systems

Transcription

1 Convex decay of Entropy for interacting systems Paolo Dai Pra Università degli Studi di Padova Cambridge March 30, 2011 Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 1 / 33

2 Based on: BOUDOU A.S., CAPUTO P., D.P., POSTA G. (2006). Spectral gap estimates for interacting particle systems via a Bochner-type identity. JOURNAL OF FUNCTIONAL ANALYSIS, vol. 232; p CAPUTO P, D.P., POSTA G (2009). Convex Entropy Decay via the Bochner-Bakry-Emery approach. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, vol. 45; p Some more recent progresses, with G. POSTA. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 2 / 33

3 Convergence to equilibrium Let π be the stationary distribution of an ergodic, continuous-time, Markov chain on a finite or countable set X, with semigroup S t = e tl. Denote by µs t the law at time t of the chain started from the law µ. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 3 / 33

4 Convergence to equilibrium Let π be the stationary distribution of an ergodic, continuous-time, Markov chain on a finite or countable set X, with semigroup S t = e tl. Denote by µs t the law at time t of the chain started from the law µ. Denoting by µ π TV := 1 µ(x) π(x) 2 x S the total variation distance, one of the aims of quantitative ergodic theory is to estimate the rate of convergence of as t +. µs t π TV 0 Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 3 / 33

5 It is usually simpler to obtain rates for other modes of convergence. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 4 / 33

6 It is usually simpler to obtain rates for other modes of convergence. For f : X R set π[f ] := x π(x)f (x). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 4 / 33

7 It is usually simpler to obtain rates for other modes of convergence. For f : X R set π[f ] := x π(x)f (x). L 2 convergence: S t π 2 2 := sup{ S t f π[f ] 2 : f 2 = 1}. where f 2 := π [f 2 ]. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 4 / 33

8 It is usually simpler to obtain rates for other modes of convergence. For f : X R set π[f ] := x π(x)f (x). L 2 convergence: S t π 2 2 := sup{ S t f π[f ] 2 : f 2 = 1}. where f 2 := π [f 2 ]. The following bound holds: letting π := min x π(x), sup δ x S t π TV 1 x S π S t π 2 2. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 4 / 33

9 Convergence in relative entropy: letting h(µ π) := ( ) µ(x) µ(x) [ µ π(x) log = π π(x) π(x) π log µ ]. π x one obtains bounds for sup h(δ x S t π). x S Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 5 / 33

10 Convergence in relative entropy: letting h(µ π) := ( ) µ(x) µ(x) [ µ π(x) log = π π(x) π(x) π log µ ]. π x one obtains bounds for sup h(δ x S t π). x S These bounds are then transferred to total variation by Czisar s inequality: sup x S h(δ x S t π) sup δ x S t π 2 TV x S Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 5 / 33

11 Functional inequalities We are therefore interested in the rate of convergence to zero of quantities as S t π 2 2 and h(δ x S t π). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 6 / 33

12 Functional inequalities We are therefore interested in the rate of convergence to zero of quantities as S t π 2 2 and h(δ x S t π). A key notion is that of Dirichlet form: E(f, g) := π [flg]. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 6 / 33

13 Functional inequalities We are therefore interested in the rate of convergence to zero of quantities as S t π 2 2 and h(δ x S t π). A key notion is that of Dirichlet form: E(f, g) := π [flg]. If the chain is reversible (i.e. π(x)l xy = π(y)l yx ) E(f, g) = 1 π [L xy (f (y) f (x))(g(y) g(x))] 2 y S (true for g = f without reversibility) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 6 / 33

14 Functional inequalities We are therefore interested in the rate of convergence to zero of quantities as S t π 2 2 and h(δ x S t π). A key notion is that of Dirichlet form: E(f, g) := π [flg]. If the chain is reversible (i.e. π(x)l xy = π(y)l yx ) E(f, g) = 1 π [L xy (f (y) f (x))(g(y) g(x))] 2 y S (true for g = f without reversibility) Also let, for f > 0 Ent π (f ) := π[f log f ] π[f ] log π[f ], so that h(µ π) = Ent π (dµ/dπ). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 6 / 33

15 Theorem S t π 2 2 e γt is equivalent to (PI) Var π [f ] := π [ f 2] π 2 [f ] 1 γ E(f, f ) for every f, which is called the Poincaré inequality. The best constant in (PI) is the spectral gap of L+L 2. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 7 / 33

16 Theorem S t π 2 2 e γt is equivalent to (PI) Var π [f ] := π [ f 2] π 2 [f ] 1 γ E(f, f ) for every f, which is called the Poincaré inequality. The best constant in (PI) is the spectral gap of L+L 2. Suppose the chain is reversible. Then h(µs t π) h(µ π)e αt for every initial distribution µ, is equivalent to (MLSI) Ent π (f ) 1 E(f, log f ), α for every f > 0. The inequality above is called the modified logarithmic Sobolev inequality. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 7 / 33

17 In particular (PI) implies max δ xs t π TV 1 x S π e γt. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 8 / 33

18 In particular (PI) implies max δ xs t π TV 1 x S π e γt. while (MLSI) and Czisar s inequality yield max δ xs t π TV x S log ( ) 1 π e αt/2 Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 8 / 33

19 In particular (PI) implies max δ xs t π TV 1 x S π e γt. while (MLSI) and Czisar s inequality yield max δ xs t π TV x S log ( ) 1 π e αt/2 It turns out that α 2γ. Thus the estimate obtained with the (MLSI) could be worse in the exponential rate, but for moderate times could be much better since, for large S, log ( ) 1 π 1 π. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 8 / 33

20 It is customary to deal with a third functional inequality, the logarithmic Sobolev inequality: for all f 0. (LSI) Ent π (f ) 1 s E( f, f ) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 9 / 33

21 It is customary to deal with a third functional inequality, the logarithmic Sobolev inequality: (LSI) Ent π (f ) 1 s E( f, f ) for all f 0. The deepest meaning of this inequality will not be dealt with here. We mention the fact that in the case of diffusion processes, i.e. when L is a second order differential operator, (MLSI) and (LSI) coincide. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 9 / 33

22 Theorem Consider an irreducible, reversible, finite state Markov chain, and let γ, α, s denote the largest constants in (PI), (MLSI) and (LSI) respectively. Then 2γ α 4s > 0 Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 10 / 33

23 Theorem Consider an irreducible, reversible, finite state Markov chain, and let γ, α, s denote the largest constants in (PI), (MLSI) and (LSI) respectively. Then 2γ α 4s > 0 For Markov chains with infinite state space the inequalities between the best constants still hold, but the constants may be zero. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 10 / 33

24 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

25 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Easy to use to get rough bounds. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

26 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Easy to use to get rough bounds. Sometimes hard to get the right scaling of the constants in terms of the parameters of the model. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

27 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Easy to use to get rough bounds. Sometimes hard to get the right scaling of the constants in terms of the parameters of the model. For model with a spatial structure, one relates functional inequalities to spatial mixing properties (Stroock, Zegarlinski, Yau, Martinelli, Olivieri,...) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

28 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Easy to use to get rough bounds. Sometimes hard to get the right scaling of the constants in terms of the parameters of the model. For model with a spatial structure, one relates functional inequalities to spatial mixing properties (Stroock, Zegarlinski, Yau, Martinelli, Olivieri,...) Quite powerful but spatial mixing can be hard to prove. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

29 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Easy to use to get rough bounds. Sometimes hard to get the right scaling of the constants in terms of the parameters of the model. For model with a spatial structure, one relates functional inequalities to spatial mixing properties (Stroock, Zegarlinski, Yau, Martinelli, Olivieri,...) Quite powerful but spatial mixing can be hard to prove. Applications to inhomogeneous models may be problematic. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

30 Main approaches to functional inequalities for Markov Chains Path (geometric) methods (Saloff-Coste, Roberto,...) Easy to use to get rough bounds. Sometimes hard to get the right scaling of the constants in terms of the parameters of the model. For model with a spatial structure, one relates functional inequalities to spatial mixing properties (Stroock, Zegarlinski, Yau, Martinelli, Olivieri,...) Quite powerful but spatial mixing can be hard to prove. Applications to inhomogeneous models may be problematic. Lyapunov function methods (Cattiaux, Guillin). Natural approach, but effectiveness on models with spatial structure is still unclear (to me) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 11 / 33

31 The Bochner-Bakry-Emery approach to PI and MLSI In this section we assume reversibility of L. We recall that the proof that the rate of L 2 convergence to equilibrium is the best constant in (PI) is based on d dt Var π(s t f ) = 2E(S t f, S t f ). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 12 / 33

32 The Bochner-Bakry-Emery approach to PI and MLSI In this section we assume reversibility of L. We recall that the proof that the rate of L 2 convergence to equilibrium is the best constant in (PI) is based on d dt Var π(s t f ) = 2E(S t f, S t f ). Taking one more derivative we get d 2 dt 2 Var π(s t f ) = 2 d dt E(S tf, S t f ) = 4π [ (LS t f ) 2]. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 12 / 33

33 The Bochner-Bakry-Emery approach to PI and MLSI In this section we assume reversibility of L. We recall that the proof that the rate of L 2 convergence to equilibrium is the best constant in (PI) is based on d dt Var π(s t f ) = 2E(S t f, S t f ). Taking one more derivative we get d 2 dt 2 Var π(s t f ) = 2 d dt E(S tf, S t f ) = 4π [ (LS t f ) 2]. Suppose now k > 0 such that for every f E(f, f ) 1 k π [ (Lf ) 2]. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 12 / 33

34 The Bochner-Bakry-Emery approach to PI and MLSI In this section we assume reversibility of L. We recall that the proof that the rate of L 2 convergence to equilibrium is the best constant in (PI) is based on d dt Var π(s t f ) = 2E(S t f, S t f ). Taking one more derivative we get d 2 dt 2 Var π(s t f ) = 2 d dt E(S tf, S t f ) = 4π [ (LS t f ) 2]. Suppose now k > 0 such that for every f E(f, f ) 1 k π [ (Lf ) 2]. We get d dt E(S tf, S t f ) 2kE(S t f, S t f ) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 12 / 33

35 In particular E(S t f, S t f ) 0 as t +. Rewriting the last inequality as d dt E(S tf, S t f ) k d dt Var π(s t f ) and integrating from t to + we get E(f, f ) kvar π (f ) k γ! Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 13 / 33

36 In particular E(S t f, S t f ) 0 as t +. Rewriting the last inequality as d dt E(S tf, S t f ) k d dt Var π(s t f ) and integrating from t to + we get E(f, f ) kvar π (f ) k γ! By a bit of spectral Theory it can be shown that the best constant k in (PI ) E(f, f ) 1 k π [ (Lf ) 2] is equal to the spectral gap γ. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 13 / 33

37 The same argument can be implemented with he entropy replacing the variance. We obtain, for f > 0 d 2 dt 2 Ent π(s t f ) = d dt E(S tf, log S t f ) [ (LSt f ) 2 ] = π [LS t fl log S t f ] + π S t f Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 14 / 33

38 The same argument can be implemented with he entropy replacing the variance. We obtain, for f > 0 d 2 dt 2 Ent π(s t f ) = d dt E(S tf, log S t f ) [ (LSt f ) 2 ] = π [LS t fl log S t f ] + π S t f We therefore have that the inequality [ ] (Lf ) (MLSI 2 ) ke(f, log f ) π [LfL log f ] + π f for every f > 0, implies the (MLSI) kent π (f ) E(f, log f ). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 14 / 33

39 The same argument can be implemented with he entropy replacing the variance. We obtain, for f > 0 d 2 dt 2 Ent π(s t f ) = d dt E(S tf, log S t f ) [ (LSt f ) 2 ] = π [LS t fl log S t f ] + π S t f We therefore have that the inequality [ ] (Lf ) (MLSI 2 ) ke(f, log f ) π [LfL log f ] + π f for every f > 0, implies the (MLSI) kent π (f ) E(f, log f ). This time the converse is not necessarily true: the entropy may decay exponentially fast, but not necessarily in a convex way. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 14 / 33

40 In order to understand how useful the above inequalities are, we write generators of Markov chains in the following form: Lf (x) = γ G c(x, γ)[f (γ(x)) f (x)] =: γ G c(x, γ) γ f (x) where G is some set of functions from X to X (allowed movements). It is clear that every countable Markov chain can be written in this way. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 15 / 33

41 In order to understand how useful the above inequalities are, we write generators of Markov chains in the following form: Lf (x) = γ G c(x, γ)[f (γ(x)) f (x)] =: γ G c(x, γ) γ f (x) where G is some set of functions from X to X (allowed movements). It is clear that every countable Markov chain can be written in this way. For L to be reversible w.r.t. a probability π we need (Rev) For every γ G there exists γ 1 G such that γ 1 γ(x) = x for every x X such that c(x, γ) > 0. Moreover π(x)c(x, γ) = π(γ(x))c(γ(x), γ 1 ) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 15 / 33

42 The inequalities (PI ) and (MLSI ) become, respectively π 4 X c(x, γ) γf (x) γf (x) 5 1 k π 4 X c(x, γ)c(x, δ) γf (x) δ f (x) 5 γ G γ,δ G and π 4 X c(x, γ) γf (x) γ log f (x) 5 γ G 2 1 k π 4 X c(x, γ)c(x, δ) γf (x) δ log f (x) + γf (x) «3 δf (x) 5 f (x) γ,δ Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 16 / 33

43 The quadratic form π c(x, γ)c(x, δ) γ f (x) δ f (x) γ,δ G can be modified as follows. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 17 / 33

44 The quadratic form π c(x, γ)c(x, δ) γ f (x) δ f (x) γ,δ G can be modified as follows. To fix ideas, set π(x) = 1 Z e H(x), c(x, γ) = exp[ γ H(x)/2], and assume γ δ = δ γ for every γ, δ. Then one shows that, for every f π c(x, γ)c(γ(x), δ) γ f (x) δ f (x) γ,δ G = 1 4 π c(x, γ)c(γ(x), δ) ( γ δ f (x)) 2 0. γ,δ G Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 17 / 33

45 Thus π 4 X c(x, γ) γf (x) γf (x) 5 γ G k π 4 X (c(x, γ)c(x, δ) c(x, γ)c(γ(x), δ)) γf (x) δ f (x) 5 γ,δ G is stronger than (PI ). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 18 / 33

46 Thus π 4 X c(x, γ) γf (x) γf (x) 5 γ G k π 4 X (c(x, γ)c(x, δ) c(x, γ)c(γ(x), δ)) γf (x) δ f (x) 5 γ,δ G is stronger than (PI ). This may be useful if c(x, γ)c(x, δ) c(x, γ)c(γ(x), δ) for many pairs (γ, δ). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 18 / 33

47 Theorem Let R : S G G [0, + ) be such that for each x, γ, δ with R(x, γ, δ) > 0 we have P1 : R(x, γ, δ) = R(x, δ, γ) P2 : π(x)r(x, γ, δ) = π(γ(x))r(γ(x), γ 1, δ) P3 : γδ(x) = δγ(x) Set Γ R (x, γ, δ) := c(x, γ)c(x, δ) R(x, γ, δ). Then we have 2 3 h π (Lf ) 2i = π 4 X c(x, γ)c(x, δ) γf (x) δ f (x) 5 γ,δ G 2 3 π 4 X Γ R (x, γ, δ) γf (x) δ f (x) 5 γ,δ G Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 19 / 33

48 The result above follows from the identity: π c(x, γ)c(x, δ) γ f (x) δ f (x) γ,δ G = π Γ R (x, γ, δ) γ f (x) δ f (x) γ,δ G + π Γ R (x, γ, δ) ( γ δ f (x)) 2 γ,δ G which is reminiscent of Bochner s identities. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 20 / 33

49 Similarly Theorem» (Lf ) 2 π [LfL log f ] + π f 2 = π 4 X c(x, γ)c(x, δ) γ,δ 2 π 4 X γ,δ γf (x) δ log f (x) + γf (x) «3 δf (x) 5 f (x) Γ R (x, γ, δ) γf (x) δ log f (x) + γf (x) «3 δf (x) 5 f (x) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 21 / 33

50 Summing up: π Γ R (x, γ, δ) γ f (x) δ f (x) ke(f, f ) γ,δ G Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 22 / 33

51 Summing up: π Γ R (x, γ, δ) γ f (x) δ f (x) ke(f, f ) and π γ,δ γ,δ G ( Γ R (x, γ, δ) γ f (x) δ log f (x) + ) γf (x) δ f (x) f (x) for every f > 0, imply (PI) and (MLSI ), respectively. ke(f, logf ) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 22 / 33

52 Glauber dynamics for unbounded particles Let Λ Z d, K, λ : Z d [0, + ) with K(0) = 0, K(x) < + λ := sup λ(x) < +. x Z d x Z d Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 23 / 33

53 Glauber dynamics for unbounded particles Let Λ Z d, K, λ : Z d [0, + ) with K(0) = 0, K(x) < + λ := sup λ(x) < +. x Z d x Z d X := N Λ. So, for η X, η x = number of particles at x Λ. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 23 / 33

54 Glauber dynamics for unbounded particles Let Λ Z d, K, λ : Z d [0, + ) with K(0) = 0, K(x) < + λ := sup λ(x) < +. x Z d x Z d X := N Λ. So, for η X, η x = number of particles at x Λ. π(η) := 1 λ(x) ηx exp β Z η x! x Λ {x,y} Λ K(x y)η x η y, Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 23 / 33

55 Glauber dynamics for unbounded particles Let Λ Z d, K, λ : Z d [0, + ) with K(0) = 0, K(x) < + λ := sup λ(x) < +. x Z d x Z d X := N Λ. So, for η X, η x = number of particles at x Λ. π(η) := 1 λ(x) ηx exp β Z η x! x Λ γ + x = creation of a particle at x. γ x = deletion of a particle at x. {x,y} Λ K(x y)η x η y, Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 23 / 33

56 Glauber dynamics for unbounded particles Let Λ Z d, K, λ : Z d [0, + ) with K(0) = 0, K(x) < + λ := sup λ(x) < +. x Z d x Z d X := N Λ. So, for η X, η x = number of particles at x Λ. π(η) := 1 λ(x) ηx exp β Z η x! x Λ γ + x = creation of a particle at x. γ x = deletion of a particle at x. {x,y} Λ K(x y)η x η y, c(η, γ x ) := η x c(η, γ x + ) := λ(x) exp β K(x y)η y y Λ Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 23 / 33

57 Choose R(η, γ x +, γ y + ) = c(η, γ y + )c(γ x + (η), γ y + ) R(η, γx, γy ) = c(η, γy )c(γx (η), γy ) R(η, γx, γ y + ) = c(η, γx )c(η, γ y + ) and write ± x for γ ± x. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 24 / 33

58 We get 2 3 h π (Lf ) 2i π 4 X Γ R (η, γ, δ) γf (η) δ f (η) 5 = X π hη x ` i x f (η) 2 γ,δ x Λ + X π ˆc(η, γ x + )c(η, γ y + ) (1 exp[ βk(x y)]) + x f (η) + y f (η) x,y Λ = E(f, f ) +... Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 25 / 33

59 We get 2 3 h π (Lf ) 2i π 4 X Γ R (η, γ, δ) γf (η) δ f (η) 5 = X π hη x ` i x f (η) 2 γ,δ x Λ + X π ˆc(η, γ x + )c(η, γ y + ) (1 exp[ βk(x y)]) + x f (η) + y f (η) x,y Λ = E(f, f ) +... Using: c(η, γ y + ) λ, 2 + x f (η) + y f (η) [ + x f (η)] 2 [ + y f (η) ] 2 we obtain Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 25 / 33

60 We get 2 3 h π (Lf ) 2i π 4 X Γ R (η, γ, δ) γf (η) δ f (η) 5 = X π hη x ` i x f (η) 2 γ,δ x Λ + X π ˆc(η, γ x + )c(η, γ y + ) (1 exp[ βk(x y)]) + x f (η) + y f (η) x,y Λ = E(f, f ) +... Using: c(η, γ y + ) λ, 2 + x f (η) + y f (η) [ + x f (η)] 2 [ + y f (η) ] 2 we obtain π [ (Lf ) 2] [1 λε(β)]e(f, f ) with ε(β) := (1 exp[ βk(x)]). x Z d Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 25 / 33

61 This proves (PI) with γ = 1 λε(β), reproving results by Bertini, Cancrini, Cesi (2002), Kondratiev & Lytvynov (2005). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 26 / 33

62 This proves (PI) with γ = 1 λε(β), reproving results by Bertini, Cancrini, Cesi (2002), Kondratiev & Lytvynov (2005). STRONGER INEQUALITIES? Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 26 / 33

63 This proves (PI) with γ = 1 λε(β), reproving results by Bertini, Cancrini, Cesi (2002), Kondratiev & Lytvynov (2005). STRONGER INEQUALITIES? (LSI) is known to fail Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 26 / 33

64 One key step was to bound the non-diagonal terms π [ c(η, γ + x )c(η, γ + y ) (1 exp[ βk(x y)]) + x f (η) + y f (η) ] using 2 + x f (η) + y f (η) [ + x f (η)] 2 [ + y f (η) ] 2. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 27 / 33

65 One key step was to bound the non-diagonal terms π [ c(η, γ + x )c(η, γ + y ) (1 exp[ βk(x y)]) + x f (η) + y f (η) ] using 2 + x f (η) + y f (η) [ + x f (η)] 2 [ + y f (η) ] 2. This is an instance of the following bound: for a symmetric matrix A = (A ij ), A D with D diagonal and d ii = a ii j i a ij. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 27 / 33

66 For the (MLSI ) we get» (Lf ) 2 π [LfL log f ] + π f 2 π 4 X Γ R (x, γ, δ) γf (x) δ log f (x) + γf (x) «3 δf (x) 5 f (x) γ,δ = E(f, log f ) + X " # π η(x) ( x f (η)) 2 f (η) x Λ + X " π c(η, γ x + )c(η, γ+ y ) 1 e βk(x y) + x f (η) + y log f (η) + + x f (η) + y f (η)!# f (η) x,y Λ Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 28 / 33

67 The role of 2ab a 2 b 2 is now played by the following inequality: Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 29 / 33

68 The role of 2ab a 2 b 2 is now played by the following inequality: (a 1) log b + (b 1) log a + 2(a 1)(b 1) (a 1)2 (b 1)2 [(a 1) log a + (b 1) log b] a b Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 29 / 33

69 The role of 2ab a 2 b 2 is now played by the following inequality: (a 1) log b + (b 1) log a + 2(a 1)(b 1) (a 1)2 (b 1)2 [(a 1) log a + (b 1) log b] a b which yields + x f (η) + y log f (η) + + y f (η) + x log f (η) x f (η) + y f (η) f (η) ` + x f (η) 2 ` + f (η + δ y f (η) 2 x) f (η + δ y ) + x f (η) + x log f (η) + y f (η) + y log f (η) Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 29 / 33

70 from which we obtain [ ] (Lf ) 2 π [LfL log f ] + π f [ [1 ε(β)] E(f, log f ) + π [η(x) ( x f (η)) 2 ] ] f (η) x Λ which means Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 30 / 33

71 from which we obtain [ ] (Lf ) 2 π [LfL log f ] + π f [ [1 ε(β)] E(f, log f ) + π [η(x) ( x f (η)) 2 ] ] f (η) x Λ which means Theorem Assuming λε(β) < 1, (MLSI ) holds with constant [1 λε(β)]. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 30 / 33

72 Some concluding remarks For the spectral gap, the methods used for bounding from below a quadratic form is very naive. Better estimates could be obtained by estimating the spectral radius of the form. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 31 / 33

73 Some concluding remarks For the spectral gap, the methods used for bounding from below a quadratic form is very naive. Better estimates could be obtained by estimating the spectral radius of the form. We worked out some examples where the naive estimates are not good enough. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 31 / 33

74 Some concluding remarks For the spectral gap, the methods used for bounding from below a quadratic form is very naive. Better estimates could be obtained by estimating the spectral radius of the form. We worked out some examples where the naive estimates are not good enough. E.g. the Loss Networks representation of Peierls contours (Fernandez, Ferrari, Garcia, 2001). This is a birth & death dynamics on Peierls contours, having the Ising model as invariant measure. We can give a uniform lower bound of the gap under the same low temperature condition as in [FFG]. Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 31 / 33

75 Some concluding remarks However, global, non-diagonal bounds are not yet available for (MLSI ). Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 32 / 33

76 THANKS! Paolo Dai Pra Convex decay of Entropy for interacting systems Cambridge, 03/11 33 / 33