Convergence to equilibrium for a directed <span class="mathjax-formula">$(1+d)-$</span>dimensional polymer

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

PIETROCAPUTO ANDJULIEN SOHIER

Convergence to equilibrium for a directed(1 +d)−dimensional polymer

Tome XXVI, n^o2 (2017), p. 289-318.

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Convergence to equilibrium for a directed (1 + d)−dimensional polymer

Pietro Caputo⁽¹⁾ and Julien Sohier⁽²⁾

ABSTRACT. — We consider a flip dynamics for directed (1+d)−dimen- sional lattice paths with lengthL. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter cor- responding to the case d = 1. We prove that the mixing time of the associated Markov chain scales likeL²logLup to ad–dependent multi- plicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scaleL²for every fixedd, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson’s argument [13] for the one-dimensional case.

RÉSUMÉ. — Nous considérons une dynamique de flips pour des che- mins de longueurLsur le réseauZ^d. Il est naturel d’interpréter ce modèle comme une généralisation multidimensionnelle du processus d’exclusion simple, qui correspond au casd= 1. Nous montrons que le temps de mé- lange de la chaîne de Markov associée se comporte commeL²logLà des constantes multiplicatives près, qui dépendent de la dimensiond. L’idée clef de la preuve pour la borne supérieure est de montrer une inégalité de Sobolev logarithmique pour une constante d’ordreL²; pour ce faire, nous combinons une récurrence sur la dimension et une estimée pour des transpositions adjacentes. Nous montrons la borne inférieure en utilisant une version de l’inégalité de Wilson [13] pour le cas unidimensionnel.

Keywords:exclusion process, adjacent transpositions, logarithmic Sobolev inequality, mixing time.

Math. classification:60K35, 82C20, 82C41.

(1) Università Roma Tre, Via della Vasca Navale, 84, 00146 Roma, Italy — caputo@mat.uniroma3.it

(2) Technische Universiteit Eindhoven, De Lampendriessen, 5612 AZ Eindhoven, The Netherlands — sohier@tue.nl

This work was partially supported by the European Research Council through the Advanced Grant PTRELSS 228032. We thank Fabio Martinelli for several stimulating discussions.

1. Introduction

Consider the set Ω_L of all Z^d paths which start and end at the origin after L steps, where L is an even integer. That is, the set of vectors η = (η0, . . . , ηL), withηx∈Z^d,η0=ηL= 0, with|ηx−ηx+1|= 1. Alternatively, we can look at ΩL as the set of all directed paths in (1 +d) dimensions which start at (o,0) and end at (o, L) where ostands for the origin of Z^d. We interpret a configuration η ∈ ΩL as a directed (1 +d)−dimensional polymer.

Consider the Markov chain where independently, at the arrival times of a Poisson clock with intensity 1, each sitex∈ {1, . . . , L−1}updates the value of η_x with a random η⁰_x chosen uniformly among all possible values of the polymer at that site given the values of the polymerη_y at all sitesy6=x. To define this process formally, letµ denote the uniform probability measure on ΩL, and write

Qxf(η) =µ(f|ηy, y6=x),

for the conditional expectation of a function f : ΩL 7→ R at x given the valuesηy at all vertices y 6= x. Then, the process introduced above is the continuous-time Markov chain with infinitesimal generator

Lf =

L−1

X

x=1

[Qxf−f], (1.1)

for all functions f : ΩL 7→ R. Note that L defines a bounded self-adjoint operator onL²(ΩL, µ). Indeed,Lis a symmetric|ΩL| × |ΩL|matrix. For any σ∈ΩL, let η_t^σ denote the polymer configuration at timet when the initial condition is σ∈ΩL, so that, for any t>0, σ, ξ ∈ΩL, the matrix element pt(σ, ξ) :=e^tL(σ, ξ) represents the probability of the eventη_t^σ=ξ. SinceLis irreducible and symmetric, one sees thatµis the unique invariant measure and

t→∞lim p_t(σ, ξ) =µ(ξ), for anyσ, ξ∈ΩL. The mixing timeTmix is defined by

Tmix = infn

t >0 : max

σ∈ΩL

kpt(σ,·)−µkTV61/4o

, (1.2)

wherek · k_TV denotes the total variation distance:

kν−ν⁰kTV= ¹₂ X

η∈ΩL

|ν(η)−ν⁰(η)|, (1.3) for probabilities ν, ν⁰ on ΩL. We refer to [11] for more background on this standard notion of mixing time.

The one-dimensional cased= 1 has been extensively studied in the past.

This model is equivalent to the simple exclusion process on the segment {1, . . . , L} with L/2 particles. It was shown by D.B. Wilson [13] that the mixing time scales like L²logL up to a multiplicative constant. More re- cently, a finer analysis of the mixing time was obtained by H. Lacoin [8], who showed that that the process exhibits a cutoff phenomenon with

T_mix = _4π¹₂ +o(1)

L²logL, (1.4)

as L → ∞. Below, we will consider the higher dimensional case d > 1, where apparently no estimates of this type have been obtained before. As explained later on, one may interpret this as a suitable exclusion process withddifferent types of particles. Our main result is as follows.

Theorem 1.1. — For any d > 2, there exist constants c, C > 0 such that the inequality

c L²logL6Tmix 6CL²logL (1.5) holds for allL.

We remark that in the special cased= 2 one can obtain the above result by using a simple product representation. Indeed, after a rotation by 45 degrees of the latticeZ²one sees that a directed (1+2)–dimensional polymer is represented as two directed (1 + 1)–dimensional polymers, the equilibrium measureµbeing the product of the two one-dimensional polymer measures.

Moreover, the dynamics is seen to coincide with the Markov chain where the two polymers are independently updated at the same random times and same random positions. These facts can be used to provide a simple proof of (1.5) by comparison with the one-dimensional case.

The general cased>3 cannot be represented in product form and thus new arguments are needed. The lower bound in (1.5) will be obtained in Sec- tion 3 below by a suitable modification of the lower bound from D.B. Wil- son [13] for the one-dimensional case. The upper bound requires more work.

A direct coupling argument does not seem to be available whend>3. An important difference with respect to the cased= 1 is the lack of standard monotonicity tools.

The main step in the proof of the upper bound is to show that the process satisfies the logarithmic Sobolev inequality with constants scaling like L². To be more precise, let

E(f, g) =−µ[fLg], (1.6) forf, g : ΩL 7→R, denote the Dirichlet form of the process, and define the entropy functional

Ent(f) =µ[flogf]−µ[f] logµ[f], (1.7)

forf : ΩL7→R+. In Section 2 we show that for anyd, for anyf : ΩL7→R+, for anyL∈2N, one has the inequality

Ent(f)6c L²E(p f ,p

f) (1.8)

wherec=c(d) is a positive constant. Once the bound (1.8) is available, the upper bound in Theorem 1.1 is obtained by an application of the standard estimates relating the constant in the log-Sobolev inequality and the mixing time.

Diffusive scaling of the constants in the log-Sobolev inequality as in (1.8) is well known to hold for the simple exclusion process and for various gener- alizations of it; see in particular [14] and [1]. However, the higher dimensional case considered here is not covered by these works. One of the main differ- ences is that the model here has dconservation laws rather than just one.

We note that if one is after the weaker Poincaré inequality, or spectral gap, then the diffusive estimate could be obtained by adapting the arguments in [2]. However, this would not suffice to prove the desired upper bound on the mixing time. To prove (1.8) instead, we exploit a recursion over the dimension such that at each step the number of particles of a new type is fixed. At the final stage of the recursion, the numbers of all particle types have been assigned, and the problem is reduced to the proof of diffusive scal- ing for the log-Sobolev constant in the setting of card shuffling by adjacent transpositions. The latter is established in Section 2.8 below. A high-level description of the whole argument is given in Section 2.2. We remark that the same argument actually proves the upper boundT_mix 6CL²logL for the more general problem where the end pointη_Lof the polymer is fixed at an arbitrary value inZ^d, not necessarily the origin, with constantCindepen- dent of the value ofη_L. For simplicity of notation we have chosen to restrict ourselves to the caseη_L=o. On the other hand it should be remarked that our argument provides a constant C = C(d) that is presumably far from optimal, especially fordlarge.

We end this introduction by mentioning an interesting open question.

Consider the above defined polymer model in the presence of a pinning po- tential, that is when µ is modified by assigning the weight λ^N(η) to each configurationη ∈ΩL, whereN(η) =PL−1

x=11^ηx=o stands for the number of contacts of the polymer with the origino ∈ Z^d, andλ > 0 is a parameter determining the strength of repulsion (λ <1) or attraction (λ >1) to the ori- gin. It is well known that the polymer undergoes a localization/delocalization phase transition, with critical pointλc(d) = 1 ford= 1,2 and λc(d)>1 for d >2; see [7, Chapter 3]. The mixing time of the polymer in the presence of pinning was studied in depth in [5] and [3] in the cased= 1, where it was shown among other things that there is a slowdown in the relaxation, with subdiffusive behavior, in the delocalized regimeλ <1. We conjecture that

this phenomenon should disappear as soon as d > 1 and that the mixing time should stay bounded byO(L²logL) for anyλ >0.

2. Proof of the upper bound.

2.1. Representation as a particle system

Recall that ΩL is the set ofZ^d paths of lengthL that start and end at the origin. For anyη∈ΩL, letζ=∇η denote the vector

ζ= (ζ1, . . . , ζL), ζx=ηx−ηx−1. (2.1) Through this map the set ΩL will be identified with the set of vectors

n

ζ∈ {e1, . . . , e2d}^L : PL

i=1ζi= 0o ,

whereej,j= 1, . . . , d, denotes the canonical basis ofZ^d, and for notational convenience we define ej+d :=−ej. For j = 1, . . . , d, and x = 1, . . . L, we say that site x is occupied by a particle of type j if ζx = ej, and by an anti-particle of typej ifζx=−ej. At each site there is either a particle or an anti-particle. Because of the constraintηL=o, for every typej= 1, . . . , d the number of particles equals the number of anti-particles. The dynamics defined by (1.1) is then interpreted naturally as a particle exchange process with creation-annihiliation of particle/anti-particle pairs as follows. Fix a polymer configurationηand letζdenote the corresponding gradient vector.

Fix a sitexto be updated. Ifη_x−16=η_x+1, then one hasζ_x=η_x−η_x−1=e_j andζ_x+1 =η_x+1−η_x=e<sub>`</sub>, for somej, `∈ {1, . . . ,2d}withe_`6=−e_j. Thus, in this case there are two possibilities for the new value ofη_x, one corresponding to the current configurationη, and one corresponding to the configuration η^xobtained by swapping the incrementsζx, ζx+1. On the other hand, if the polymer is such thatη_x−1=ηx+1, then one must have (ζx, ζx+1) = (ej,−ej), for somej∈ {1, . . . ,2d}. Thus, in this case there are 2dpossibilities for the new value ofηx. We callη^x,∗,j the polymer configuration that coincides with ηat all sitesy6=xand such that (ζx, ζx+1) = (ej,−ej). Thus, in this process adjacent particles exchange their positions, and when a particle/anti-particle pair occupies adjacent sites it can be deleted to produce a new particle/anti- particle pair of a different type. With this notation, the generator (1.1) takes the form

Lf(η) =

L−1

X

x=1

cx(η)[f(η^x)−f(η)] +

L−1

X

x=1 2d

X

j=1

c^?,j_x (η)[f(η^x,?,j)−f(η)], (2.2) where

cx(η) := ¹₂1^ηx−16=ηx+1, c^?,j_x (η) := _2d¹ 1^ηx−1=η_x+1.

The Dirichlet form (1.6) of this process becomes E(f, g) =¹₂

L−1

X

x=1

µ[c_x(∇xf)²] +¹₂

L−1

X

x=1 2d

X

j=1

µ[c^?,j_x (∇^∗,j_x f)²], (2.3) where we use the notation∇xf(η) =f(η^x)−f(η), and∇^∗,j_x f(η) =f(η^x,?,j)−

f(η). Notice that whend= 1 only the last term in the right hand side of (2.3) survives and we obtain the symmetric simple exclusion process; see e.g. [5].

2.2. Overview of the proof

Let us describe the strategy for the upper bound of Theorem 1.1. Recall the definition of the Dirichlet form (1.6) and define the log-Sobolev constant

α(L) = inf

f

E(√ f ,√

f)

Ent(f) , (2.4)

wheref ranges over all functionsf : Ω_L7→R+.

We refer to [6] for the following classical inequality relatingα(L) toT_mix: T_mix 64 + log(log(1/π^?))

2α(L) , (2.5)

withπ^? := minx∈Ω_Lµ(x) =|ΩL|⁻¹. Since|ΩL|6(2d)^L, it suffices to prove the following estimate.

Theorem 2.1. — For anyd>1, there exist a constant c=c(d)>0:

α(L)>c L⁻². (2.6)

It is worth remarking that the estimate (2.6) is sharp up to a constant factor, as one sees using (2.5) and the lower bound onT_mixfrom Theorem 1.1.

In the special cased= 1, corresponding to the exclusion process, the above theorem was known before; see [14]. We now illustrate the main steps of the proof.

LetN_i denote the number of particles of typei:

N_i(η) =

L

X

x=1

1ζx=ei, (2.7)

where as aboveζ_x =η_x−η_x−1. The law of the vector (N₁, . . . , N_d) under the uniform distributionµis given by

µ(N1=n1, . . . , Nd=nd) = 1

|ΩL| L

L/2

L/2 n1, . . . , nd

2

(2.8)

where (n₁, . . . , n_d) is a vector of non negative integers satisfyingPd j=1n_j= L/2, and _n^L/2

1,...,n_d

is the associated multinomial coefficient. Indeed, (2.8) follows easily by counting all possible choices of positions ofni particles of typeiandni anti-particles of typei, fori= 1, . . . , d.

We denote byµ⁽ⁱ⁾,i= 1, . . . , d, the measureµ(· |N₁, . . . , N_i) obtained by conditioningµ on a given value of the numbers of particles of type 1, . . . , i, and we write µ⁽⁰⁾ = µ. Notice that µ^(d−1) = µ^(d) because of the global constraintPd

j=1N_j=L/2. We write

Enti(f) =µ⁽ⁱ⁾[flogf]−µ⁽ⁱ⁾[f] logµ⁽ⁱ⁾[f],

for the entropy with respect to the measureµ⁽ⁱ⁾. Thus, Enti(f) is a function of the variablesN1, . . . , Ni.

Roughly speaking, the proof proceeds by induction fromi=dto i= 0.

In the base case i = d, all numbers N_i are fixed and the only degree of freedom is the position of the particles. In this case the dynamics reduces to adjacent swaps, which can be analyzed in terms of the interchange process;

see Theorem 2.3 below. To move fromi+1 toiwe first decompose the entropy Enti(f) along the random variableNi+1. We then estimate the log-Sobolev constant of a birth and death dynamics for the variable Ni+1. Finally, a delicate comparison argument allows one to recast the estimate for the birth and death chain in terms of the Dirichlet form of the original process plus an error term that can be absorbed in the recursion; see Theorem 2.2 below.

We turn to the details. By adding and subtractingµ[flogµ(f|X)] in the expression (1.7) one obtains the following standard decomposition of entropy via conditioning on a random variableX:

Ent(f) = Ent(µ(f|X)) +µ(Ent(f|X)), (2.9) where Ent(f|X) is the entropy off with respect to the measure conditioned on the value of X and µ(f|X) is the X–measurable function defined by conditional expectation. Applied to the measure µ⁽ⁱ⁾ with X = N_i+1, and noting thatµ⁽ⁱ⁾(f|Ni+1) =µ⁽ⁱ⁺¹⁾(f), (2.9) gives that for any 06i6d−1, Enti(f) = Enti(µ⁽ⁱ⁺¹⁾(f)) +µ⁽ⁱ⁾[Enti+1(f)]. (2.10) Sinceµ^(d−1)=µ^(d) we note that Entd−1(µ^(d)(f)) = 0. A crucial step in our proof will be the following estimate.

Theorem 2.2. — For any d>2, there exist constants c1, c2 >0 such that for alli= 0, . . . , d−2:

µ(Ent_i(µ⁽ⁱ⁺¹⁾(f)))6c₁L²E(p f ,p

f) +c₂µ(Ent_i+1(f)), (2.11) for allf : ΩL 7→R+.

Once this result is available we proceed as follows. For i = 0, (2.10) and (2.11) give the estimate

Ent(f)6c1L²E(p f ,p

f) + (1 +c2)µ(Ent1(f)).

If we iterate this reasoning, we obtain the estimate Ent(f)6k1L²E(p

f ,p

f) +k2µ(Entd(f)), (2.12) where k₁ = c₁Pd−1

i=0(1 +c₂)ⁱ and k₂ = (1 +c₂)^d. It remains to estimate µ(Ent_d(f)).

Theorem 2.3. — There exists a constantC >0 such that µ(Ent_d(f))6CL²E(p

f ,p

f), (2.13)

for allf : ΩL 7→R+.

Theorem 2.2 and Theorem 2.3 then allow us to conclude that the log- Sobolev constant in (2.4) satisfies

α(L)⁻¹6(C k2+k1)L²,

which ends the proof of Theorem 2.1. The following subsections are devoted to the proof of Theorem 2.2 and Theorem 2.3.

2.3. Log-Sobolev inequality for a birth and death chain

The starting point in the proof of Theorem 2.2 is an application of a criterion for log-Sobolev inequalities in birth and death chains due to L. Mi- clo [12]. For this purpose we follow [1]. In the definition below, we con- sider a generic probability measure γ on the finite set of integers S :=

{nmin, nmin+ 1, . . . , nmax}, for somenmax> nmin.

Definition 2.4 (Condition Conv(c,n))¯ . — We say that γ satisfies the convexity hypothesis with parameters c > 0 and n¯ ∈ S, which we denote by Conv(c,n), if¯ c⁻¹n¯ 6nmax−n¯ 6c¯n and the same inequality holds for

¯

n−nmin. Furthermore, for any n>¯n:

γ(n+ 1)

γ(n) 6c e^{−n−¯c¯nn}, (2.14) and for anyn6n,¯

γ(n−1)

γ(n) 6c e^{−n−n¯c¯n} . (2.15) Finally for anyn∈S,

1 c√

¯

ne^{−c( ¯n−n)2¯n} 6γ(n)6 c

√¯ne^{−( ¯n−n)2c¯n} . (2.16)

The following useful lemma appears in [1]. We use the notationa∧b= min{a, b}.

Lemma 2.5 ([1, Proposition A.5]). — Ifγ satisfies Conv(c,n), then for¯ all functionsg:S 7→R+,

Ent_γ(g)6Cn¯

nmax

X

n=nmin+1

[γ(n)∧γ(n−1)]p

g(n)−p

g(n−1)²

, (2.17) where the constantC depends only on c and not on ¯n.

We shall prove that the number of particles of a given type has a distri- bution with the properties described above. Fix i ∈ {0, . . . , d−1}, and fix nonnegative integers n1, . . . , ni such that Pi

j=1nj 6 L/2. Set S = {0, . . . , Li+1/2}, where we defineLi+1 :=L−2Pi

j=1nj. Letγ denote the probability onS:

γ(n) :=µ(Ni+1=n|N1=n1, . . . , Ni=ni) =µ⁽ⁱ⁾(Ni+1=n). (2.18) As an immediate corollary of Proposition A.1 in the appendix (simply replace LbyLi anddbyd−ithere), we have that the measureγdefined in (2.18) satisfies Conv(c,n) for some absolute constant¯ c >0, with ¯n:=Li+1/2(d− i). Therefore, for any f : ΩL 7→ R+, it follows from Proposition A.1 and Lemma 2.5 applied tog(n) =µ⁽ⁱ⁾(f|Ni+1=n), that

Enti(µ⁽ⁱ⁺¹⁾(f)) 6 C L_i+1

2(d−i)

Li+1/2

X

n=1

[γ(n)∧γ(n−1)]ρ

µ⁽ⁱ⁾(f|n), µ⁽ⁱ⁾(f|n−1)

, (2.19) whereC >0 is a constant and we use the notation

µ⁽ⁱ⁾(f|n) :=µ⁽ⁱ⁾(f|Ni+1=n) and

ρ(a, b) := √ a−√

b²

, a, b>0. (2.20) To proceed towards the proof of Theorem 2.2 we now estimate the right hand side of (2.19).

2.4. Decomposition of ρ µ⁽ⁱ⁾(f|n), µ⁽ⁱ⁾(f|n−1) .

We need to introduce some more notation. Supposeu, v∈ {1, . . . , L}and η∈ΩL is such thatζ_u=−ζv =e_j, for somej∈ {1, . . . ,2d}, where ζ=∇η as in (2.1). For any`∈ {1, . . . ,2d}, we define the operatorT<sub>u,v</sub><sup>∗,`</sup> by

T_u,v^{∗,`</sup>f(η) =f(η<sup>∗,`}_u,v), (2.21)

whereη_u,v<sup>∗,`</sup> denotes the configurationη<sup>0</sup> ∈ΩL that is equal toη except that the pair (ζu, ζv) = (ej,−ej) has been replaced by the pair (ζ<sub>u</sub><sup>0</sup>, ζ<sub>v</sub><sup>0</sup>) = (e`,−e`).

Notice that this operation is well defined, producing a valid element of ΩL, whenever the configuration η satisfies ζu = −ζv. Moreover, for any fixed u, v∈ {1, . . . , L}, anyi∈ {0, . . . , d−2}, and`∈Ai :={i+ 2, . . . , d} ∪ {d+ i+ 2, . . . ,2d}, n∈ {1, . . . , Li+1/2}, from the uniformity of µwe obtain the change of variable formula:

µ⁽ⁱ⁾ f1^ζu=−ζv=e_i+11^Ni+1=n

=µ⁽ⁱ⁾ T_u,v^∗,i+1f1^ζu=−ζv=e_`1^Ni+1=n−1 , (2.22) for any functionf. SincePL

u,v=11^ζu=−ζv=e_i+1 =N_i+1² one has µ⁽ⁱ⁾(f|n) = 1

γ(n)µ⁽ⁱ⁾

f1Ni+1=n

= 1

n²γ(n)

L

X

u,v=1

µ⁽ⁱ⁾

f1^ζu=−ζ_v=e_i+11^Ni+1=n

= 1

2(d−i−1)n²γ(n)

L

X

u,v=1

X

`∈A_i

µ⁽ⁱ⁾

T_u,v^∗,i+1f1^ζu=−ζv=e_`1^Ni+1=n−1

= γ(n−1) 2(d−i−1)n²γ(n)

L

X

u,v=1

X

`∈Ai

µ⁽ⁱ⁾

T_u,v^∗,i+1f1ζu=−ζv=e`|N_i+1=n−1 . (2.23) We introduce the notation

χu,v,`= γ(n−1)

2(d−i−1)n²γ(n)1^ζu=−ζv=e_`, χ=

L

X

u,v=1

X

`∈Ai

χu,v,`,

(2.24)

so that (2.23) takes the more compact form µ⁽ⁱ⁾(f|n) = X

u,v,`

µ⁽ⁱ⁾ χ_u,v,`T_u,v^∗,i+1f|n−1

. (2.25)

Considering the constant functionf ≡1 one has the normalization µ⁽ⁱ⁾(χ|n−1) = X

u,v,`

µ⁽ⁱ⁾(χu,v,`|n−1) = 1. (2.26) Moreover, using symmetry, for any`∈A_i we have

µ⁽ⁱ⁾ N_`²|n−1

= n²γ(n)

γ(n−1). (2.27)

From (2.25), using the inequalityρ(a, b)62ρ(a, c)+2ρ(b, c), valid fora, b, c>

0, one has ρ

µ⁽ⁱ⁾(f|n), µ⁽ⁱ⁾(f|n−1) 62ρ

P

u,v,`µ<sup>(i)</sup> χ<sub>u,v,`</sub>T_u,v^∗,i+1f|n−1

, µ⁽ⁱ⁾(χf|n−1) + 2ρ

µ⁽ⁱ⁾(χf|n−1), µ⁽ⁱ⁾(f|n−1)

=:A_i(f, n) +B_i(f, n). (2.28)

The contributions of the two terms above to the expression (2.19) will be analyzed separately.

2.5. Estimating P

n[γ(n−1)∧γ(n)]Ai(f, n).

Here we prove the following estimate.

Proposition 2.6. — There exists a constantC >0such that for allL∈ 2N,i= 0, . . . , d−1, for all even L_i+1 ∈[2, L], all integersn∈[1, L_i+1/2], and for all functionsf : Ω_L 7→R+, one has

L_i+1X

n

[γ(n−1)∧γ(n)]A_i(f, n)

6CL²

L−1

X

x=1

µ⁽ⁱ⁾[c_x(∇_xp

f)²] +CL

L−1

X

x=1 2d

X

j=1

µ⁽ⁱ⁾[c^?,j_x (∇^∗,j_x p

f)²]. (2.29) Proof. — Sinceρ: [0,∞)²7→Ris convex, by Jensen’s inequality and the expressions (2.24) and (2.26),

Ai(f, n)62X

u,v,`

µ⁽ⁱ⁾ ρ T_u,v^∗,i+1f, f

χ_u,v,`|n−1

. (2.30)

We now turn to the estimate of the right hand side of (2.30). We need to compare the exchanges at positions u, v with local exchanges between adjacent positions. Fixu, v and assume without loss of generality that v>

u+ 1. Seth=√

f so that

ρ T_u,v^∗,i+1f, f

= T_u,v^∗,i+1h−h² .

The operationT_u,v^∗,i+1 can be implemented by first transferring ζu from po- sitionuto positionv−1, through a chain of adjacent swaps, then applying the operationT_v−1,v^∗,i+1 and then finally transferring back the new value ofζ_v−1 from positionv−1 to positionuvia adjacent swaps. This can be formalized as follows. LetTudenote the adjacent swap operator that changes (ζu, ζu+1)

into (ζ_u+1, ζ_u), that isT_uh(η) =h(η^u) for any functionh: ΩL7→R; see (2.2).

Thus for any functionhwe can write

T_u,v^∗,i+1h=Tu◦. . .◦T_v−2◦T_v−1,v^∗,i+1◦T_v−2◦. . .◦Tuh.

In terms of the gradient operators∇u =Tu−1,∇^∗,i+1_u =T_u^∗,i+1−1, one has the telescopic sum

T_u,v^∗,i+1h−h

=

v−u−3

X

j=0

∇u+jT_u+j+1◦. . .◦T_v−2◦T_v−1,v^∗,i+1◦T_v−2◦. . .◦T_uh

+∇v−2T_v−1,v^∗,i+1◦Tv−2◦. . .◦Tuh+∇^∗,i+1_v−1 Tv−2◦. . .◦Tuh +

v−u−2

X

j=1

∇u+jTu+j−1◦. . .◦Tuh+∇uh . (2.31)

By the uniformity ofµ⁽ⁱ⁾, every term in the first sum in (2.31) satisfies

µ⁽ⁱ⁾

∇u+jT_u+j−1◦. . .◦T_v−2◦T_v−1,v^∗,i+1◦T_v−2◦. . .◦Tuh2

χu,v,`|n−1

= γ(n) γ(n−1)µ⁽ⁱ⁾h

(∇u+jh)²χu+j−1,v,i+1|ni

. (2.32) The same identity holds for the first term in the second line of (2.31), i.e.

whenu+j=v−2. In a similar way, for the terms in the last line of (2.31), one obtains

µ⁽ⁱ⁾h

(∇u+jT_u+j−1◦. . .◦Tuh)²χu,v,`|n−1i

=µ⁽ⁱ⁾h

(∇u+jh)²χ_u+j−1,v,`|n−1i

. (2.33) Finally, for the term involving the gradient∇^∗,i+1_v−1 one has

µ⁽ⁱ⁾

∇^∗,i+1_v−1 T_v−2◦. . .◦Tuh²

χu,v,`|n−1

=µ⁽ⁱ⁾

∇^∗,i+1_v−1 h²

χv−1,v,`|n−1

. (2.34)

From (2.30)–(2.34), using Cauchy–Schwarz inequality we have

Ai(f, n)66L²

L−1

X

x=1 L

X

v=1

X

`∈Ai

( γ(n) γ(n−1)µ⁽ⁱ⁾h

(∇xh)²χ_x−1,v,i+1|ni

+µ⁽ⁱ⁾h

(∇xh)²χ_x−1,v,`|n−1i )

+ 6L

L−1

X

x=1

X

`∈Ai

µ⁽ⁱ⁾h

∇^∗,i+1_x h²

χ_x,x+1,`|n−1i

. (2.35)

Recalling (2.24), one has that for anyx:

γ(n) γ(n−1)

L

X

v=1

X

`∈Ai

χ_x−1,v,i+16 N_i+1 n² ,

L

X

v=1

X

`∈Ai

χx−1,v,` 6γ(n−1) n²γ(n) Li+1,

X

`∈A_i

χx,x+1,`6γ(n−1)

n²γ(n) 1^ζx=−ζx+1.

Next, we claim that Li+1

n [γ(n−1)∧γ(n)] 6Cγ(n) (2.36) L²_i+1

n² [γ(n−1)∧γ(n)]γ(n−1)

γ(n) 6Cγ(n−1), (2.37) for some constantC >0. Sincen6L_i+1it is clear that (2.37) implies (2.36).

On the other hand (2.37) follows from

minn 1

n²,γ(n−1) n²γ(n)

o

6 C

L²_i+1,

which is an immediate consequence of the estimate γ(n−1)/(n²γ(n)) 6 C(Li+1−2n)⁻²; see Lemma A.2 in the appendix (applied withL replaced byLi+1 andd replaced byd−i). For the first term in (2.35), using (2.36)

we then obtain Li+1

L_i+1/2

X

n=1

(γ(n)∧γ(n−1))

L−1

X

x=1 L

X

v=1

X

`∈Ai

γ(n) γ(n−1)µ⁽ⁱ⁾h

(∇xh)²χ_x−1,v,i+1|ni

6

L_i+1/2

X

n=1

L_i+1

n [γ(n−1)∧γ(n)]

L−1

X

x=1

µ⁽ⁱ⁾h

(∇xh)² |ni

6C

Li+1/2

X

n=1 L−1

X

x=1

γ(n)µ⁽ⁱ⁾h

(∇xh)² |ni 6C

L−1

X

x=1

µ⁽ⁱ⁾h

(∇xh)²i .

For the second term in (2.35), using (2.37) we have L_i+1

L_i+1/2

X

n=1

(γ(n)∧γ(n−1))

L−1

X

x=1 L

X

v=1

X

`∈Ai

µ⁽ⁱ⁾h

(∇xh)²χ_x−1,v,`|n−1i

6

Li+1/2

X

n=1

L²_i+1

n² [γ(n−1)∧γ(n)]γ(n−1) γ(n) µ⁽ⁱ⁾h

(∇xh)²|n−1i

6C

L_i+1/2

X

n=1 L−1

X

x=1

γ(n−1)µ⁽ⁱ⁾h

(∇_xh)²|n−1i 6C

L−1

X

x=1

µ⁽ⁱ⁾h

(∇_xh)²i .

Similarly, the last term in (2.35) satisfies L_i+1

L_i+1/2

X

n=1

(γ(n)∧γ(n−1))

L−1

X

x=1

X

`∈Ai

µ⁽ⁱ⁾h

∇^∗,i+1_x h²

χ_x,x+1,`|n−1i

6C

L−1

X

x=1

µ⁽ⁱ⁾h

∇^∗,i+1_x h²i .

This ends the proof of Proposition 2.6.

2.6. Covariance estimate

Here we estimate the contribution of the second term in (2.28).

Proposition 2.7. — There exists a constantC >0such that for allL∈ 2N,i= 0, . . . , d−2, for all even Li+1 ∈[2, L], all integersn∈[1, Li+1/2], and for all functionsf : ΩL 7→R+, one has

Li+1

X

n

[γ(n−1)∧γ(n)]Bi(f, n)6C µ⁽ⁱ⁾(Enti+1(f)). (2.38)

Proof. — Note that we may assume thati6d−3 here since otherwise the function χ in (2.24) is deterministically equal to 1 under the measure µ⁽ⁱ⁾[· |n−1], and thereforeBi(f, n) = 0 for allnandf. Using the inequality

√x−√ y2

6(x−y)²

x∨y 6 (x−y)²

y ,

valid forx, y>0, one has

B_i(f, n)62µ⁽ⁱ⁾(f(χ−1)|n−1)²

µ⁽ⁱ⁾(f|n−1) . (2.39) Since by (2.26)χsatisfiesµ⁽ⁱ⁾(χ|n−1) = 1, one has

µ⁽ⁱ⁾(f(χ−1)|n−1) = Cov_i(f, χ|n−1),

where Covi(· |n−1) denotes covariance with respect toµ⁽ⁱ⁾(· |n−1). Let us define

D(n, L_i+1) = 1 Li+1

1∨γ(n−1) γ(n)

. We are going to prove that for some constantC >0 one has

Cov_i(f, χ|n−1)²6CD(n, L_i+1)µ⁽ⁱ⁾(f|n−1) Ent_i(f|n−1), (2.40) where Enti(· |n−1) stands for the entropy with respect toµ⁽ⁱ⁾(· |n−1).

If (2.40) holds, then (2.39) implies

Li+1[γ(n−1)∧γ(n)]Bi(f, n)62Cγ(n−1) Enti(f|n−1).

Using

X

n

γ(n−1) Enti(f|n−1) =µ⁽ⁱ⁾(Enti+1(f)),

we obtain the desired inequality (2.38). Thus, it suffices to prove (2.40).

To prove (2.40), by homogeneity, we may assume without loss of gener- ality thatµ⁽ⁱ⁾(f|n−1) = 1. In Proposition 2.8 below we establish that for some constantC1>0 one has the Laplace transform bound

logµ⁽ⁱ⁾

e^t(χ−1)|n−1

6C₁t²D(n, L_i+1), t∈R. (2.41) We remark that (2.40) follows easily from (2.41). Indeed, set for simplicity ν := µ⁽ⁱ⁾(· |n−1) and write Ent_ν(·) for the corresponding entropy. The variational principle for entropy implies that for any f > 0 with ν(f) = 1 one has

Entν(f) =ν(flogf)>ν(f h)−logν(e^h), for any functionh. Therefore

ν(f(χ−1))6 ¹s Entν(f) +¹_slogν

e^s(χ−1)

6¹s Entν(f) +C₁sD,

for alls >0, where we write D=D(n, L_i+1) and we use (2.41) witht=s.

Using also (2.41) witht=−sone concludes that

|ν(f(χ−1))|6¹s Entν(f) +C1sD, for alls >0. Settings=p

D⁻¹Ent_ν(f) one obtains ν(f(χ−1))²6(1 +C1)²DEntν(f),

which is the desired estimate (2.40). It remains to prove (2.41).

2.7. Laplace transform estimate

Here we prove (2.41).

Proposition 2.8. — There exists a constant C1 >0 such that (2.41) holds for allt∈R.

Proof. — We use as above the shorthand notationν=µ⁽ⁱ⁾(· |n−1) and D=D(n, L_i+1). From (2.24)–(2.27) we have

χ−1 = Pd

j=i+2(N_j²−ν[N_j²]) (d−i−1)ν[N_i+2² ] . Define the centered variables ¯N_j :=N_j−ν[N_j], where

ν[N_j] = L_i+1−2(n−1) 2(d−i−1) . Observe that by the conservation laws one has

d

X

j=i+2

N_j = L_i+1−2(n−1) 2 and thereforePd

j=i+2N¯_j= 0. From these relations we see that χ−1 = 1

d−i−1

d

X

j=i+2

Yj, Yj :=

N¯_j²−σ² ν[N_j²] ,

where we define the variance σ² :=ν[ ¯N_j²]. Using the multi-index Hölder’s inequality:

ν

e^t(χ−1) 6

d

Y

j=i+2

ν e^{t Yj}

1

d−i−1 =ν e^{t Y} ,

where the last identity follows from the fact that allY_j have the same dis- tribution under ν, sayY :=Y_i+2. Thus, it is sufficient to prove that there exists some constantC >0 such that for allt∈Rone has

logν e^{t Y}

6C t²D. (2.42)

The proof of (2.42) is divided into several steps, corresponding to different sets of values for the parameterst andn.

For simplicity, we only consider the case t > 0. The case t 60 follows with the very same arguments. We often write C, C1, C2, . . . for positive constants that are independent of the parameters n, Li+1, L etc. but may depend ond. Their value may change from line to line.

From Lemma A.3 in the appendix we know that σ² is proportional to (Li+1−2n). Notice that

ν[N_i+2² ]>ν[N_i+2]²=_L

i+1−2(n−1) 2(d−i−1)

²

, (2.43)

and thatN_i+2² 6(Li+1−2(n−1))². Therefore

Y 64(d−i−1)². (2.44)

Suppose thatt>aD⁻¹for a fixed constanta >0. Then ν e^tY

6e^4d2t6e^4d2t2D/a, which implies (2.42) withC= 4d²/a.

Next, assume thatt6b for some fixed constantb >0. From (2.43) and Lemma A.3 applied to the variableX = ( ¯N_i+2)²−σ² in the system of size L_i+1−2(n−1), we have that

Varν(Y)6C(Li+1−2n)⁻².

From Lemma A.2 applied to the system of sizeLi+1, one has that D>c(Li+1−2n)⁻¹,

for some positive constant c. Combining these facts with the well known inequality

ν(e^h)6exp ¹₂ν[h²e^|h|] ,

which is valid for any functionhwithν(h) = 0 (usee^a61 +a+¹₂a²e^|a|and 1 +x6e^x), we get

ν e^tY

6exp (t²C1(Li+1−2n)⁻¹e^4d2b)6e^C2Dt2.

Thus, we have shown that (2.42) holds for allt 6b and t>aD⁻¹, and we have freedom on the choices ofaandb. In particular, we can considerasmall andblarge if we wish.

Next, we observe that (2.42) holds for allt>0 if (L_i+1−2n)6p L_i+1. Indeed, from Lemma A.2, we know that D>c > 0 for some c >0 in this case. Therefore, taking suitable constants a, b (that is a small and b large enough) we cover allt>0 with the previous argument.

Thus, we are left with the case (L_i+1−2n)>p

L_i+1 for allt∈[b, aD⁻¹].

Since by Lemma A.2 one hasD⁻¹6C(Li+1−2n) for some constantC >0, we may actually restrict tot∈[b, c(Li+1−2n)] wherec can be made small if we wish.

Lemma 2.9. — There exists a constant c >0 such that for all n satis- fyingLi+1−2n>p

Li+1, for allt6c(Li+1−2n)we have c ν e^tY

61. (2.45)

Proof. — We compute ν e^tY

=X

k

ν( ¯Ni+2=k) exp t_ν[N^k2−σ2

i+2]²

. (2.46)

Usingt6c(Li+1−2n) andν[N_i+2² ]>(Li+1−2n)²/4d², see (2.43), we can bound

exp t_ν[N^k2−σ2

i+2]²

6exp

4d²c k² Li+1−2n

. From Proposition A.1 we know that

ν( ¯N_i+2=k)6 C

pL_i+1−2nexp

−_C(L ^k2

i+1−2n)

(2.47)

for some constantC >0. Thus, takingcsmall enough, one has that (2.46) is bounded by a constant. Adjusting the value of constants yields the desired

conclusion (2.45).

An immediate consequence of Lemma 2.9 is that (2.42) holds for allt∈ [(Li+1−2n)^1/2, c(Li+1−2n)]. Indeed, it suffices to observe that here

logν e^tY

6C₁6C₁t²/(Li+1−2n)6Ct²D, for some new constantC >0.

Therefore, for the proof of (2.42) we are left with the regimet∈[b,(L_i+1− 2n)^1/2], and (Li+1−2n)>p

Li+1. Here we use the following two facts.

Lemma 2.10. — For anyδ >0, there exists a constantc1>0 such that for all n satisfying Li+1−2n>p

Li+1 and for allt 6c1(Li+1−2n) one has

X

|k|>(L_i+1−2n)^1/2+δ

ν( ¯Ni+2=k) exp t_ν[N^k2−σ2

i+2]²

6exp −c1(Li+1−2n)^δ (2.48)