Hong-Quan Tran
email: tran@ceremade.dauphine.fr

The mean-field Zero-Range process with unbounded monotone rates: mixing time, cutoff, and Poincaré constant

We consider the mean-field Zero-Range process in the regime where the potential function r is increasing to infinity at sublinear speed, and the density of particles is bounded. We determine the mixing time of the system, and establish cutoff. We also prove that the Poincaré constant is bounded away from zero and infinity. This mean-field estimate extends to arbitrary geometries via a comparison argument. Our proof uses the path-coupling method of Bubley and Dyer and stochastic calculus.

Introduction 1.Model

The Zero-Range process, introduced by Spitzer, is a model of interacting particle systems in continuous time. It describes the evolution of m ≥ 1 indistinguishable particles jumping randomly across n ≥ 1 sites, where the speed of a particle only depends on the number of its cooccupants (hence the name Zero-Range). More precisely, the interaction is represented by a function r : {1, 2, ...} → (0, ∞), called the potential function, where r(k) is the rate at which a site with k particles expels a particle. For convenience, we let r(0) = 0 (no jump from empty sites). In this paper, we focus on the mean-field version of the model, where a jumping particle chooses its destination uniformly among all sites. More precisely, we consider a continuous-time Markov chain X := (X(t)) t≥0 = (X 1 (t), X 2 (t), ..., X n (t)) t≥0 taking values in the state space

Ω := x = (x 1 , x 2 , ..., x n ) ∈ Z n + : n i=1 x i = m , (1)
whose Markov generator L acts on an observable φ : Ω → R as follows:

(Lφ)(x) = 1 n 1≤i,j≤n r(x i )(φ(x -δ i + δ j ) -φ(x)). (2)
Here (δ i ) 1≤i≤n denotes the canonical basis of Z n + . The generator L is irreducible and reversible with respect to the following law:

π(x) ∝ n i=1 x i k=1 1 r(k) , (3)
with the convention that an empty product is 1. The classical theory of Markov processes says that starting from any probability on the state space Ω, X(t) will converge in distribution to the stationary law π as t → ∞. The speed of convergence from the state x ∈ Ω is quantified by the so-called mixing times: t mix (x; ϵ) := min{t ≥ 0 : d tv P t x , π ≤ ϵ}.

Here d tv (•, •) is the total variation distance, defined by d tv (µ, ν) = max A⊂Ω |µ(A) -ν(A)|, and P t x denotes the distribution of X(t) under the probability P x : P t x (•) = P x [X(t) ∈ •], where P x is the law of the process starting from x. Of particular interest is the worst-case mixing time: t mix (ϵ) := max{t mix (x; ϵ) : x ∈ Ω}, [START_REF] Hermon | A version of Aldous' spectral-gap conjecture for the zero range process[END_REF] where we take the maximum of the mixing times over all initial configurations x.

We recall that the Dirichlet form associated with our process is defined by:

E(φ, ψ) := -⟨φ, Lψ⟩ π ,
where ⟨φ, ψ⟩ π := x∈Ω π(x)φ(x)ψ(x) denotes the usual inner-product in L 2 (Ω, π). Then the Poincaré constant, denoted by λ * , is defined by:

λ * := min E(φ, φ) Var [φ] ,
where the minimum is taken over all non-constant observables, and Var [φ] denotes the variance of φ under π. In our case, λ * coincides with the more classical absolute spectral gap due to reversibility of the system:

λ * = lim t→∞ - 1 t log max x∈Ω d tv P t x , π .
The purpose of the present paper is to estimate λ * and t mix (x; ϵ), under certain assumptions on r(•).

Previous works. To the best of our knowledge, the total-variation mixing time of the Zero-Range process has only been studied in a few cases: the case where r is constant in [START_REF] Lacoin | The cutoff profile for the simple exclusion process on the circle[END_REF], [START_REF] Lacoin | Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion[END_REF], [START_REF] Merle | Cutoff for the mean-field zero-range process[END_REF], the case where r is non-decreasing and bounded in [START_REF] Hermon | Cutoff for the mean-field zero-range process with bounded monotone rates[END_REF], and the somehow-trivial case of independent walkers where r is linear. Regarding the Poincaré constant, a notable result is given in [START_REF] Morris | Spectral gap for the zero range process with constant rate[END_REF], where Morris determines the order of magnitude of λ * in the case where r is constant. Another result is obtained by Caputo in [START_REF] Caputo | Spectral gap inequalities in product spaces with conservation laws[END_REF] for the case where r is homogeneously Lipschitz and increasing at infinity, i.e.

sup k≥1 |r(k + 1) -r(k)| < ∞, (6)
inf k-l≥δ r(k) -r(l) > 0, [START_REF] Lacoin | The cutoff profile for the simple exclusion process on the circle[END_REF] for some δ ∈ Z + , where he proves that the Poincaré constant is bounded away from zero. In [START_REF] Hermon | A version of Aldous' spectral-gap conjecture for the zero range process[END_REF], Salez and Hermon prove a comparison principle that allows us to compare the Poincaré constant of many models with that of the mean-field model. We will use this principle below.

Main results

We consider the "intermediate" regime where the function r is non-decreasing, unbounded but grows slower than a linear function. More precisely, throughout the paper, we assume that r satisfies:

r(k + 1) ≥ r(k), ∀k ∈ Z + , (8)
lim k→∞ r(k) = ∞, (9)
sup

k∈Z + r(k) k < ∞. (10)
We study the regime where the number of sites diverges while the density of particles per site remains bounded. More precisely, we always suppose that m = m(n) and x = x (n) , and all asymptotic statements refer to the regime:

n → ∞, m n ≤ ρ, ( 11
)
where ρ is a positive constant. To lighten the notation, we keep the dependency upon n implicit as much as possible. By a dimension-free constant, we mean a number that depends only on r and ρ. Our notation O (•) (resp. Ω(•), Θ(•), o(•)) means being upper bounded by (resp. lower bounded by, upper and lower bounded by, negligible compared to) the quantity inside the brackets up to a dimension-free prefactor. We define a function R : {1, 2, ...} → R as follows:

∀k ∈ Z + , R(k) = k i=1 1 r(i) . ( 12
)
Under condition [START_REF] Merle | Cutoff for the mean-field zero-range process[END_REF], we easily see that

lim k→∞ R(k) = ∞. (13)
We prove that R(∥x∥ ∞ ) is a good estimate for t mix (x; ϵ), as stated in the following theorem:

Theorem 1 (Main result). For ϵ ∈ (0, 1) fixed, for any initial state x,

t mix (x; ϵ) ≤ (1 + o(1))R(∥x∥ ∞ ) + O (log n) . (14)
In addition, if the initial state

x = x (n) satisfies x (n) ∞ n→∞ ---→ ∞, then t mix (x; ϵ) ≥ (1 -o(1))R(∥x∥ ∞ ).
Maximizing over all initial states x, we obtain Corollary 2 (Cutoff). Suppose additionally that R(m) ≫ log n. Then for ϵ ∈ (0, 1) fixed,

t mix (ϵ) R(m) = 1 + o(1). ( 15
)
In other words, the system exhibits cutoff at time R(m).

The class of functions r that satisfy conditions ( 8), ( 9), ( 10) is quite large. A natural example is when r is of the form r(k) = k α , ∀k ∈ Z + , for some α ∈ (0, 1). In this case,

R(k) = (1 + o(1)) k 1-α 1 -α , as k → ∞,
by the Stolz-Cesàro Theorem. Thereupon, a direct application of our result gives the following.

Example 1. Suppose that r(k) = k α , ∀k ∈ Z, for some α ∈ (0, 1), and suppose that m ≫ (log n) 1/(1-α) . Then the system exhibits cutoff at time m 1-α 1 -α .

Cutoff for the Zero-Range process was obtained in [START_REF] Merle | Cutoff for the mean-field zero-range process[END_REF] for the case r(k) = 1 and more generally in [START_REF] Hermon | Cutoff for the mean-field zero-range process with bounded monotone rates[END_REF] for the case where r is non-decreasing and bounded. Our work complements these results by investigating the case where r → ∞. We also prove that the Poincaré constant is bounded away from zero and infinity:

Theorem 3 (Poincaré constant). λ * = Θ(1).
Thanks to the comparisons in the paper [START_REF] Hermon | A version of Aldous' spectral-gap conjecture for the zero range process[END_REF] of Hermon and Salez, we can extend this result to the more general case where a jumping particle chooses its destination according to a doubly stochastic matrix P rather than uniformly among all sites (for example, take P to be the transition matrix of random walk on a regular graph). More precisely, let P be an irreducible doubly stochastic transition matrix on [n] := {1, 2, ..., n}, and let L P be the generator on Ω that acts on an observable φ : Ω → R by:

(L P φ)(x) = 1≤i,j≤n r(x i )P (i, j)(φ(x -δ i + δ j ) -φ(x)).
Similarly, we can define the Poincaré constants λ * (P ) and λ * (L P ) of P and L P via their associated Dirichlet forms and their stationary laws. Then we have the following.

Corollary 4 (Poincaré constant in arbitrary geometry). λ * (L P ) = Θ(λ * (P )).

We give an example where we can compute λ * (P ) explicitly to obtain explicit estimate on λ * (L P ).

Example 2 (Poincaré constant of torus model). Let P be the transition matrix of the simple random walk on the lattice Z d /pZ d , for some p, d ∈ Z + . Then λ * (L P ) = Θ(1/(dp 2 )), as p d → ∞.

The calculation of λ * (P ) is defered to the end of the paper.

Heuristics. If we ignore arrivals and only consider departures of particles, then R(∥x∥ ∞ ) is exactly the expectation of the time it takes for the initially highest site to be emptied. In the true system, due to the conditions imposed on r, the arrival rate at each site is uniformly bounded, and consequently, R(∥x∥ ∞ ) remains a good approximation for the emptying-time. For the lower bound, we prove that before time R(∥x∥ ∞ ), the initially highest site still has too many particles, and hence the system has not yet reached equilibrium. For the upper bound, we will see that at time

t = (1 + o(1))R(∥x∥ ∞ ) + O (log n), ∥X(t)∥ ∞ = O (log n).
Afterwards, the system quickly reaches equilibrium.

Acknowledgment. The author warmly thanks Justin Salez for constructive discussions and his comments on the draft. The author also kindly thanks the anonymous referee for his suggestion to make the paper more clear and readable.

2 Lower bound on the mixing time

Preliminaries

We will use the following two graphical constructions of the process X.

Graphical construction 1. Let Ξ be a Poisson point process of intensity

1 n dt⊗du⊗Card⊗Card on [0, ∞) × [0, ∞) × [n] × [n],
where Card denotes the counting measure. Define the piece-wise constant process X = (X(t)) t≥0 taking values in Ω as follows: X(0) = x, and for each point (t, u, i, j) of Ξ,

X(t) := X(t-) -δ i + δ j , if u ≤ r(X i (t-)) X(t-) otherwise. ( 16
)
Then X is a càdlàg Markov process starting from x with generator L.

Graphical construction 2. Let Ψ be a Poisson point process of intensity dt

⊗ du ⊗ Card on [0, ∞) × [0, ∞) × [n]
. Consider the piece-wise constant process X = (X(t)) t≥0 which starts at X(0) = x and has the following jumps: for each point (t, u, j) of Ψ,

X(t) :=      X(t-) -δ i + δ j , if 1 n i-1 k=1 r (X k (t-)) < u ≤ 1 n i k=1 r(X k (t-)), for some i ∈ [n] X(t-) otherwise. ( 17
)
Then X is also a Markov process starting from x with generator L. We can view the Poisson process in the graphical construction 2 as the repartition of the Poisson process in the graphical construction 1 according to the destination of the jumps.

Filtration. We always note (F t ) t≥0 the filtration generated by the Poisson processes in the graphical construction we are using, where F t is the σ -algebra generated by these processes up to time t. It is immediate that the process X is adapted to the filtration and has càdlàg trajectories.

Mean-field jump rate. At any time t, as the model is mean-field, the arrival rate at each site is the same. We denote this quantity by ζ(t):

ζ(t) := 1 n n j=1 r(X j (t)).
Condition [START_REF] Merle | Cutoff for the mean-field zero-range process[END_REF] implies that

ζ(t) ≤   1 n n j=1 X j (t)   sup k∈Z + r(k) k ≤ ρ sup k∈Z + r(k) k =: κ. (18)
Hence, the number of particles arriving at each site is stochastically dominated by a Poisson process of dimension-free intensity κ.

Martingale associated with an observable. For any observable φ : Ω → R, under P x , the process M = (M (t)) t≥0 given by

M (t) := φ(X(t)) -φ(x) - t 0 Lφ(X(u))du (19)
is a zero-mean martingale, see e.g [START_REF] Ethier | Markov processes -characterization and convergence[END_REF]. Let φ 1 , φ 2 be two observables, and let M 1 , M 2 be the associated martingales. Then the predictable covariation of M 1 and M 2 is given by

⟨M 1 , M 2 ⟩ t = t 0 y∈Ω L(X(u), y) (φ 1 (y) -φ 1 (X(u))) (φ 2 (y) -φ 2 (X(u))) du. (20)
We recall a lemma on the concentration of martingales with jumps (see [START_REF] Van De Geer | Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes[END_REF]):

Lemma 5 (Concentration of martingale). Let (M (t)) t≥0 be a zero-mean càdlàg martingale w.r.t a filtration that satisfies the usual conditions. Suppose that M (0) = 0 and M (t) -M (t-) ≤ K for all t > 0 and some 0 ≤ K < ∞. Then for each a > 0, b > 0,

P ∃t ≥ 0 : M (t) ≥ a, ⟨M, M ⟩ t ≤ b 2 ≤ exp - a 2 2(aK + b 2 ) . ( 21
)
Gain/loss at a site. We will need the following quantities:

1. For i ∈ [n], let G i (t) be the counting process that counts the number of particles arriving at site i up to time t. We call G i the gain at site i.

2. Let L i (t) be the counting process that counts the number of particles jumping out of site i up to time t. We call L i the loss at site i.

Obviously, X i (t) = X i (0) + G i (t) -L i (t).

Elementary concentration inequalities

We write ξ ∼ exp(λ) to mean that ξ is an exponential variable with parameter λ, i.e. ξ has density λe -λx 1 {x>0} dx. We list here some useful inequalities, whose proofs are simple applications of Chernoff's bound (see, e.g. [START_REF] Boucheron | Concentration inequalities: A nonasymptotic theory of independence[END_REF] for more details on Chernoff's bound).

Lemma 6 (Poisson concentration). Let Z be a Poisson variable. Then for any B > 1,

P [Z ≥ BEZ] ≤ e -(1+B ln B-B)EZ .
Lemma 7 (Concentration of sum of independent exponential variables). Let λ 1 , ..., λ k be positive numbers, and let ξ 1 , ..., ξ k be independent random variables such that

ξ i ∼ exp(λ i ), ∀1 ≤ i ≤ k. Let S := k i=1 ξ i .
For B > 0 arbitrary, we have the following inequalities:

P S -ES ≤ -Var [S]B ≤ e -B 2 /4 , (22)
and

P S -ES ≥ λVar [S] + B λ ≤ e -B/2 , (23)
where λ = min 1≤i≤k {λ i }.

Proof of the lower bound

In this subsection, we prove the lower bound on t mix (x; ϵ) in Theorem 1. First, we analyze the law of a single site at equilibrium.

Proposition 8 (Single site marginals at equilibrium).

There exists a dimension-free constant q > 0 such that

∀k ∈ Z + , π(x 1 = k) π(x 1 = k -1) < q r(k) .
Proof. Let x be an arbitrary configuration in Ω such that x 1 ≥ 1. Thanks to [START_REF] Morris | Spectral gap for the zero range process with constant rate[END_REF], the number of sites that have at least 2ρ particles is at most n 2 , and hence the number of sites that have less than

2ρ particles is at least n 2 . For any l ∈ [n] \ {1} such that x l < 2ρ, thanks to (8), π(x) π(x -δ 1 + δ l ) = r(x l + 1) r(x 1 ) ≤ r(⌈2ρ⌉) r(x 1 )
.

Taking average over all sites l such that x l < 2ρ, we obtain

π(x) ≤ r(⌈2ρ⌉) r(x 1 ) 2 n l̸ =1 x l <2ρ π(x -δ 1 + δ l ) ≤ r(⌈2ρ⌉) r(x 1 ) 2 n l̸ =1 π(x -δ 1 + δ l ).
Now we take the sum over all x such that x 1 = k to get:

π(x 1 = k) ≤ r(⌈2ρ⌉) r(k) 2 n x 1 =k l̸ =1 π(x -δ 1 + δ l ) = r(⌈2ρ⌉) r(k) 2 n l̸ =1 x 1 =k π(x -δ 1 + δ l ) = r(⌈2ρ⌉) r(k) 2 n l̸ =1 z 1 =k-1, z l ≥1 π(z) (change of variable: z = x -δ 1 + δ l ) ≤ r(⌈2ρ⌉) r(k) 2 n l̸ =1 z 1 =k-1 π(z) = r(⌈2ρ⌉) r(k) 2 n l̸ =1 π(x 1 = k -1) ≤ r(⌈2ρ⌉) r(k) 2n n π(x 1 = k -1) = q r(k) π(x 1 = k -1),
where q = 2r(⌈2ρ⌉).

Now we study the effect of arrivals at a particular site. Recall that L i denotes the loss at site i, as defined at the end of subsection 2.1.

Lemma 9 (Effect of arrivals). Let x be an arbitrary initial configuration. For i ∈ [n] and h

∈ Z + such that x i ≥ h, there exist independent variables U k ∼ exp n -1 n r(k) , h ≥ k ≥ 1, such that for any 0 ≤ k ≤ h -1, 1. T k := U h + U h-1 + ... + U h-k is a stopping time, 2. almost surely, L i (T k ) ≥ k + 1 , 3. almost surely, X i (t) ≥ h -k, ∀t ∈ [0, T k ).
The intuition is as follows: if we ignore arrivals and consider only departures at site i, then T k is the time to have k + 1 particles depart from i. Arrivals, on the one hand, slow down a site from being emptied, but on the other hand, accelerate the rate of expelling and hence increase the loss. So the inequalities at points 2 and 3 follow.

Proof. We use the graphical construction 1. We first prove for the case k = 0. Let U h be defined by

U h = inf t ≥ 0 : Ξ [0, t] × [0, r(h)] × {i} × [n] \ {i} > 0 . It is clear that U h is a stopping time and U h ∼ exp n -1 n r(h) . Moreover, by definition of U h , Ξ [0, U h ) × [0, r(h)] × {i} × [n] \ {i} = 0, so before time U h , X i cannot fall from h to h -1. In other words, ∀t ∈ [0, U h ), X i (t) ≥ h. In particular, X i (U h -) ≥ h, so there should be a jump from site i to [n] \ {i} at U h . Consequently, L i (U h ) ≥ 1 and X i (U h ) = X i (U h -) -1 ≥ h -1,
almost surely, which finishes the case k = 0. Now we define U k inductively by:

U k = inf t ≥ 0 : Ξ (T h-k-1 , T h-k-1 + t] × [0, r(k)] × {i} × [n] \ {i} > 0 ,
where

T h-k-1 = U h + ... + U k+1 .
The variables (U k ) h≥k≥1 are independent by the stationary and independent increments of Poisson processes. In addition,

U k ∼ exp n -1 n r(k) .
The claim is simply obtained by induction and by the strong Markov property.

Useful variables. Lemma 9 allows us to compare certain random times with the random variables (S k ) k≥1 defined by

S k = k i=1 ξ i , (24)
where (ξ i ) i≥1 is a sequence of independent random variables such that ξ i ∼ exp(r(i)), ∀i ∈ Z + .

Obviously, 2 . Due to (8), ( 9), [START_REF] Merle | Cutoff for the mean-field zero-range process[END_REF], the functions r, R diverge, so we easily see that

E [S k ] = k i=1 1 r(i) = R(k), Var [S k ] = k i=1 1 r(i)
lim k→∞ Var [S k ] ES k = 0. ( 25
)
Lemma 9 makes precise the fact that arrivals can only slow down a site from being emptied, while Proposition 8 together with condition (9) say that at equilibrium, the typical height of a site cannot be very large. This leads to the lower bound on t mix (x; ϵ) in Theorem 1:

Proof of the lower bound in Theorem 1. Let δ ∈ (0, 1) be fixed, and let x ∈ Ω be arbitrary. We only need to prove that for ∥x∥ ∞ sufficiently large,

t mix (x; ϵ) R(∥x∥ ∞ ) > 1 -δ. ( 26
)
Without loss of generality, suppose that site 1 is originally the highest, i.e. ∥x∥ ∞ = x 1 . We know that for any A ⊂ Ω,

d tv P t x , π ≥ P t (x, A) -π(A). We choose A = {y ∈ Ω : y 1 ≥ k},
where

k = sup l ∈ Z + : |R(l)| ≤ δ 2 R(∥x∥ ∞ ) .
We only need to show that π(A) = o(1), and for t = (1 -δ)R(x 1 ), P t x (A) = 1 -o(1). Thanks to [START_REF] Van De Geer | Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes[END_REF], ∥x∥ ∞ ≫ 1 ensures that R(∥x∥ ∞ ) ≫ 1 and hence k ≫ 1, so by Proposition 8 and (9), π(A) = o(1). On the other hand, we apply Lemma 9 with i = 1, h = x 1 to conclude that there exists a stopping time

T x 1 -k-1 (d) = n n -1 (S x 1 -S k ),
where the sequence (S k ) k≥1 is defined in (24), such that

X 1 (t) ≥ k + 1, ∀t ∈ [0, T x 1 -k-1
). We define S k,x 1 = S x 1 -S k , for any k < x 1 . Then

P (1-δ)R(x 1 ) x [A] ≥ P x [T x 1 -k-1 > (1 -δ)R(x 1 )] ≥ P [S k,x 1 > (1 -δ)R(x 1 )] = 1 -P [S k,x 1 ≤ (1 -δ)R(x 1 )] . Moreover, R(k) ≤ δ 2 R(x 1
) by definition of k, and hence

E [S k,x 1 ] = R(x 1 ) -R(k) ≥ (1 -δ/2)R(x 1 ).
So by the concentration inequality (22),

P [S k,x 1 ≤ (1 -δ)R(x 1 )] ≤ P S k,x 1 -ES k,x 1 ≤ - δ 2 R(x 1 ) ≤ exp - 1 4 • δ 2 4 R(x 1 ) 2 Var [S k,x 1 ] -1 ≤ exp - δ 2 16 R(x 1 )Var [S x 1 ] -1 . = o(1),
where in the third inequality we have used the fact that Var [S x 1 ] > Var [S k,x 1 ], and in the last equality we have used (25) and the fact that

x 1 = ∥x∥ ∞ n→∞ ---→ ∞. Hence P t x (A) = 1 -o(1)
, which finishes our proof.

Upper bound on the mixing time

We will prove the following statements: Proposition 10 (Dissolution). There exist dimension-free constants σ, α 1 such that for any δ ∈ (0, 1) fixed, for any x ∈ Ω, for any t

≥ (1 + δ)R(∥x∥ ∞ ) + σ log n, P x [∥X(t)∥ ∞ ≥ α 1 log n] = O n -2 .
Proposition 11 (Quick convergence to equilibrium). Let α 1 be defined as in Proposition 10. Then there exists a dimension-free constant α so that for any configuration x such that ∥x∥ ∞ ≤ α 1 log n,

d tv P α log n x , π = O n -2 .
First we see how these propositions lead to the upper bound on t mix (x; ϵ) in Theorem 1:

Proof of the upper bound in Theorem 1: Let σ, α 1 , α be defined as in Proposition 10 and Proposition 11. Let x be an arbitrary configuration; let t 1 = (1 + δ)R(∥x∥ ∞ ) + σ log n, for some δ, t 2 = α log n, and t = t 1 + t 2 . We only need to prove that for arbitrary δ ∈ (0, 1) fixed, for n large enough, for any x ∈ Ω, t mix (x; ϵ) ≤ t.

By the convexity of the total variation distance,

d tv P t x , π ≤ y∈Ω P t 1 x (y)d tv P t 2 y , π ≤ P x [∥X(t 1 )∥ ∞ ≥ α 1 log n] + max ∥y∥ ∞ ≤α 1 log n d tv P t 2 y , π ,
which is O n -2 by Proposition 10 and Proposition 11, hence smaller than ϵ when n is large enough.

The structure of the rest of this section is as follows. In Subsection 3.1, we prove Proposition 10 and provide some analysis on the trajectory of the system which will be used in the proof of Proposition 11. In Subsection 3.2, we prove Proposition 11 by the path coupling method of Bubley and Dyer.

Dissolution

Recall that G i denotes the gain at site i, as defined at the end of subsection 2.1. First we give an estimate on G i at the predicted time.

Lemma 12 (Estimating the gain at a site). For any dimension-free constant d, there exists a dimension-free constant c 0 such that at time t = (1 + δ/4)(R(∥x∥ ∞ ) + d log n), for any δ ∈ (0, 1)

fixed, P [G i (t) ≥ c 0 (R(∥x∥ ∞ ) ∨ log n)] = O n -5 . ( 27
)
Proof. We use the graphical construction 2. Since ζ(t) < κ at all time (see (18)), G i is simply dominated by the Poisson process Ψ

•×[0,κ]×{i} . Consequently, at time t = (1 + δ/4)(R(∥x∥ ∞ ) + d log n), G i (t) is stochastically dominated by a random variable Y ∼ Poisson κ(1+δ/4)(R(∥x∥ ∞ )+ d log n) .
Then the result is simply obtained by Lemma 6, for c 0 large enough.

For any site i, Lemma 9 says that arrivals can only accelerate the loss L i , while Lemma 12 gives us a (random) upper bound on G i . They together lead to the following proposition:

Proposition 13 (First phase dissolution). Let d = 7 r(1)
, and let c 0 be defined as in Lemma 12.

Then for any x ∈ Ω, for any δ ∈ (0, 1) fixed, for t

= (1 + δ/4)(R(∥x∥ ∞ ) + d log n), P x [∥X(t)∥ ∞ ≤ c 0 (R(∥x∥ ∞ ) ∨ log n)] = 1 -O n -2 . ( 28
) Proof. Let i ∈ [n]
. We apply Lemma 9 with h = x i to conclude that there exists a stopping time

T x i -1 (d) = n n -1 S x i such that L i (T x i -1 ) ≥ x i . Note that S x i is dominated stochastically by S ∥x∥ ∞ ,
hence by using (23), we deduce that

P T x i -1 ≥ n n -1 E S ∥x∥ ∞ + r(1)Var S ∥x∥ ∞ + 6 log n r(1) = P S x i ≥ E S ∥x∥ ∞ + r(1)Var S ∥x∥ ∞ + 6 log n r (1)
≤ P S ∥x∥ ∞ ≥ E S ∥x∥ ∞ + r(1)Var S ∥x∥ ∞ + 6 log n r (1)
≤ n -3 . 25), and hence

Note that Var

S ∥x∥ ∞ = o(R(∥x∥ ∞ ) ∨ log n) by (
n n -1 E S ∥x∥ ∞ + r(1)Var S ∥x∥ ∞ + 6 log n r(1) < t,
when n is large enough. Consequently, for n large enough,

P x [L i (t) < x i ] ≤ P x [T x i -1 > t] ≤ n -3 .
We take a union bound of this and the event in ( 27) over all sites to conclude that

P [∃i, G i (t) ≥ c 0 (R(∥x∥ ∞ ) ∨ log n)] + P [∃i, L i (t) < x i ] = O n -2 .
The claim follows.

We now recall a simple version of Gronwall's lemma that we will use a lot:

Lemma 14 (Gronwall's lemma). Let α, β be some positive numbers. Let u : [0, ∞) → R + be a continuously differentiable function such that d dt u(t) ≤ -βu(t) + α. Then

u(t) < α β + u(0) - α β e -βt .
In particular, if t ≥ log u(0) β , then u(t) < α β + 1.

For θ a positive number, and for i ∈ [n], we define the observable

φ θ i : Ω → R by φ θ i (x) = e θx i , (29)
and we define the observable φ θ by

φ θ (x) = 1 n n i=1 φ θ i (x).
Lemma 15 (Estimate on Lφ θ ). For any dimension-free constants θ, β > 0, there exists a number L = L(θ, β) such that, for any configuration x,

Lφ θ (x) ≤ -βφ θ (x)1 {φ θ (x)>L} + (e θ -1)κφ θ (x)1 {φ θ (x)≤L} . ( 30
)
In particular,

Lφ θ (x) ≤ -βφ θ (x) + ((e θ -1)κ + β)L. (31)
Proof. For simplicity, we write φ instead of φ θ and φ i instead of φ θ i . It is not difficult to see that

Lφ i (x) φ i (x) = e θ -1 n j∈[n]\{i} r(x j ) - 1 -e -θ n j∈[n]\{i} r(x i ) = e θ -1 n j∈[n] r(x j ) -(1 -e -θ ) n -1 n + e θ -1 n r(x i ).
Hence, by (18),

Lφ i (x) ≤ (e θ -1)κφ i (x) -(1 -e -θ )φ i (x)r(x i ).
Taking the average over all sites i we get

Lφ(x) ≤ (e θ -1)κφ(x) -(1 -e -θ ) i∈[n] r(x i )φ i (x) n .
The claim follows when φ(x) ≤ L. It remains to consider the case φ(x) > L. For any c ∈ Z + , r(x i )φ i (x) ≥ r(c)(φ i (x) -e θc ) due to the monotonicity of r, hence

i∈[n] r(x i )φ i (x) ≥ r(c) i∈[n] (φ i (x) -e θc ) ≥ r(c)(nφ(x) -ne θc ).
Consequently,

Lφ(x) ≤ (e θ -1)κφ(x) -(1 -e -θ )r(c)(φ(x) -e θc ).
Let L = L(c) = 2e θc . If φ(x) > L, then φ(x) -e θc > φ(x) 2 , which implies:

Lφ(x) ≤ (e θ -1)κφ(x) -(1 -e -θ )r(c) φ(x) 2 . ( 32
)
We can take c large enough to make the right-hand side of the inequality above smaller than -βφ(x), which finishes the proof of (30). ( 31) is obtained by rewriting (30) as follows,

Lφ(x) ≤ -βφ(x)(1 -1 {φ(x)≤L} ) + (e θ -1)κφ(x)1 {φ(x)≤L} = -βφ(x) + ((e θ -1)κ + β)φ(x)1 {φ(x)≤L} ≤ -βφ(x) + ((e θ -1)κ + β)L,
which is what we want.

A good estimate on Lφ θ will guarantee good behavior of the trajectories of X, as stated in the following proposition.

Proposition 16 (Dissolution). Let θ, β > 0 be some dimension-free constants. Then for any x ∈ Ω,

for any t ≥ θ β ∥x∥ ∞ , P x ∥X(t)∥ ∞ ≥ 6 θ log n = O n -5 . (33)
Proof. We still write φ instead of φ θ , for simplicity. Let L = L(θ, β) be defined as in Lemma 15.

Let

u(t) = E x [φ(X(t))]. By (19), d dt u(t) = E x [Lφ(X(t))] .
This and (31) imply:

d dt u(t) ≤ -βu(t) + ((e θ -1)κ + β)L.
Therefore, by Lemma 14,

u(t) ≤ ((e θ -1)κ + β)L β + u(0)e -βt . Hence for t ≥ θ β ∥x∥ ∞ , u(t) ≤ ((e θ -1)κ + β)L β +1. Note that e θ∥x∥ ∞ ≤ nφ(x), hence E e θ∥X(t)∥ ∞ ≤ nu(t).
Then the claim is a simple consequence of Chernoff's bound.

We now prove Proposition 10:

Proof of Proposition 10. We fix δ ∈ (0, 1). Let c 0 be defined as in Lemma 12. Let θ 1 > 0 be fixed, and let β 1 be such that

θ 1 c 0 β 1 ≤ δ 2 . Let L 1 = L(θ 1 , β 1 ) as in Lemma 15. Let t 1 = (1 + δ/4)(R(∥x∥ ∞ ) + d log n), t 2 = δ 2 (R(∥x∥ ∞ ) ∨ log n)
. By definition of β 1 and Proposition 13,

P x θ 1 ∥X(t 1 )∥ ∞ β 1 ≤ t 2 = 1 -O n -2 .
By Markov property at time t 1 and inequality (33) , we conclude that for any t ≥ t 1 + t 2 ,

P x ∥X(t)∥ ∞ ≥ 6 θ 1 log n = O n -2 .
We choose

α 1 = 6 θ 1 and σ = 5 4 d + 1 2
to conclude the proof.

The estimate on Lφ θ in Lemma 15 also ensures that the system quickly reaches the set where φ θ is small. Proposition 17 (Exponential moment of hitting time). Let θ, β > 0 be some dimension-free constants, and let L = L(θ, β) be defined as in Lemma 15. Let T be the hitting time of the set {x ∈ Ω : φ θ (x) ≤ L}. Then for any x ∈ Ω,

E x e βT ≤ e θ φ θ (x)/L. ( 34
)
Proof. We write φ instead of φ θ . The idea is that if φ is large, then the drift Lφ is negative, and its magnitude is of the same order as φ. Hence the system will quickly reach the set where φ is small, which will be made precise by stochastic calculus. Consider the function F : R 2 → R defined by F (u, v) = uv, which is twice continuously differentiable. By (19),

φ(X(t)) = φ(X(0)) + t 0 Lφ(X(u))du + M (t),
where M (t) is a martingale. Moreover, φ(X) is a pure-jump process since X is piece-wise constant.

For a càdlàg process Y , we denote by ∆Y its jumps: ∆Y (s) = Y (s) -Y (s-). We define G(t) = F (e βt , φ(X(t))). Note that the function t → e βt has bounded variation. Consequently, applying Itô's formula (for example, see Theorem 33 in chapter 2 of [START_REF] Protter | Stochastic integration and differential equations[END_REF]) to the function F and the semimartingales t → φ(X(t)) and t → e βt , we get:

G(t) =φ(X(0)) + t 0 e βu Lφ(X(u))du + t 0 e βu dM (u) + t 0 φ(X(u))βe βu du + 0≤s≤t ∆G(s) - ∂F ∂v e βs-, φ(X(s-)) • ∆φ(X(s)) . (35)
On the other hand, as ∂F ∂v (u, v) = u and the function t → e βt is continuous, ∆G(s) = e βs φ(X(s)) -e βs φ(X(s-)) = e βs ∆φ(X(s)) = ∂F ∂v e βs-, φ(X(s-)) • ∆φ(X(s)).

Hence the last term in the right-hand side of (35) is zero. Moreover, the term t 0 e βu dM (u) is a martingale. Applying the formula at time t ∧ T , we get: (36) By Lemma 15, Lφ(X(u)) + βφ(X(u)) ≤ 0 when u < T . It follows that e β(t∧T ) φ(X(t ∧ T )) is a supermartingale. Thus,

e β(t∧T
E x e β(t∧T ) φ(X(t ∧ T )) ≤ φ(x). (37)
Clearly, if x ∈ arg min φ, then Lφ(x) ≥ 0, and hence by (30), φ(x) ≤ L. In particular, {φ ≤ L} ̸ = ∅, so T < ∞ a.s. as the process is irreducible. In (37), letting t → ∞ and using Fatou's lemma, we obtain

E x e βT φ(X(T )) ≤ φ(x).
It is easy to see that φ(X(T )) ≥ φ(X(T -))/e θ ≥ Le -θ , which leads to our claim.

The next lemma says that if we start from a configuration x such that ∥x∥ ∞ = O (log n), then this remains true for a long time.

Lemma 18 (Stability of trajectories). Let α 1 be defined as in Proposition 10. There is a dimensionfree constant α 2 such that

sup ∥x∥ ∞ ≤α 1 log n P x ∃t ∈ [0, (log n) 2 ], ∥X(t)∥ ∞ > α 2 log n = O n -3 . (38) Proof. Suppose that ∥x∥ ∞ ≤ α 1 log n. Let θ 1 = 6 α 1
, and let β 1 be a constant, and let t =

θ 1 β 1 α 1 log n ≥ θ 1 β 1 ∥x∥ ∞ .
Then by (33),

P x [∥X(t)∥ ∞ > α 1 log n] = O n -5 .
Moreover, by Lemma 6, for a dimension-free constant α ′ 2 large enough,

P G i (t) ≥ α ′ 2 log n = O n -5 .
Taking a union bound, we deduce that

P x {∥X(t)∥ ∞ > α 1 log n} ∪ {∃i : G i (t) ≥ α ′ 2 log n} = O n -4 .
This implies

P x ∥X(t)∥ ∞ ≤ α 1 log n, sup s∈[0,t] ∥X(s)∥ ∞ ≤ (α 1 + α ′ 2 ) log n ≥ 1 -O n -4 .
The inequality remains true when we take the supremum over all x such that ∥x∥ ∞ ≤ α 1 log n. Iterating, and using the Markov property, we deduce that, for any k ∈ Z + ,

P x ∥X(kt)∥ ∞ ≤ α 1 log n, sup s∈[0,kt] ∥X(s)∥ ∞ ≤ (α 1 + α ′ 2 ) log n ≥ (1 -O n -4 ) k ≥ 1 -O kn -4 .
We finish the proof simply by taking k = ⌊(log n) 2 ⌋, and

α 2 = α 1 + α ′ 2 .
In the next proposition, we prove that the bound on ∥X∥ ∞ above leads to a strong bound on φ θ (X) for some θ > 0.

Proposition 19 (Strong concentration of trajectories). Let θ 2 and β 2 be two positive dimensionfree constants and L 2 = L(θ 2 , β 2 ) as in Lemma 15. Let T be the hitting time of the set {φ θ 2 ≤ L 2 }. Let α 1 be defined as in Proposition 10. Then, provided that θ 2 is small enough, for any x such that ∥x∥ ∞ ≤ α 1 log n,

P x sup s∈[T,(log n) 2 ] φ θ 2 (X(s)) > L 2 + 4 = O n -3 . ( 39
)
We will need the following lemma:

Lemma 20 (Martingale estimate). Let α 1 be defined as in Proposition 10. Let θ 2 be a positive dimension free constant. Suppose that ∥x∥ ∞ ≤ α 1 log n, and let (M (t)) t≥0 be defined by

M (t) = φ θ 2 (X(t)) -φ θ 2 (X(0)) - t 0 Lφ θ 2 (X(u))du,
which is a martingale according to (19). Then, for θ 2 small enough,

P x sup s∈[0,(log n) 2 ] |M (s)| ≥ 1 = O n -3 .
For simplicity, in the proofs of Lemma 20 and Proposition 19, we still write φ instead of φ θ 2 and φ i instead of φ θ 2 i . First we see how Lemma 20 leads to Proposition 19:

Proof of Proposition 19. Let θ 2 and M be as in Lemma 20. We will prove that

sup s∈[T,(log n) 2 ] φ(X(s)) > L 2 + 4 ⊂ sup s∈[0,(log n) 2 ] |M (s)| ≥ 1 T ≥ (log n) 2 ,
and then we show that the probabilities of the events on the right-hand side is O n -3 . By contrapositivity, suppose that for a realization of X which is càdlàg almost surely, we have sup

s∈[0,(log n) 2 ] |M (s)| < 1 and T < (log n) 2 . We prove that sup s∈[T,(log n) 2 ] φ(X(s)) ≤ L 2 + 4. For h ∈ [T, (log n) 2 ] arbitrary, let s h = sup{s ∈ [0, h] : φ(X(s-)) ≤ L 2 }. Note that for any s ∈ [0, (log n) 2 ], |∆φ(X(s))| = |∆M (s)| ≤ 2 sup s∈[0,(log n) 2 ] |M (s)| ≤ 2.
Moreover, by definition of s h , φ(X(s h -)) ≤ L 2 , and hence φ(X(s h ))

≤ L 2 + ∆φ(X(s h )) ≤ L 2 + 2.
Also by definition of s h , φ(X(u)) > L 2 when s h ≤ u < h, and hence Lφ(X(u)) < 0 by Lemma 15. Consequently,

φ(X(h)) = φ(X(s h )) -M (s h ) + M (h) + h s h Lφ(X(u))du ≤ L 2 + 4, ( 40
)
which proves the inclusion. Besides, Proposition 17 gives us P x T ≥ (log n) 2 = O n -3 by Chernoff's bound. Combining this and Lemma 20, we deduce the claim.

Now we prove Lemma 20:

Proof of Lemma 20. We will provide good control on ∆M and ⟨M, M ⟩, and afterward we use Lemma 5. By (19), the process M i (t) defined by

M i (t) = φ i (X(t)) -φ i (X(0)) - t 0 Lφ i (X(u))du
is a zero-mean martingale. It it clear that M is the average of M i :

M (t) = 1 n n i=1 M i (t).
Due to the conservation of the number of particles, the martingales (M i ) i∈[n] have negative covariations. More precisely, according to (20), we have

d ⟨M i , M j ⟩ t = 1≤k,l≤n r(X k (t)) n φ i (X(t) -δ k + δ l ) -φ i (X(t)) φ j (X(t) -δ k + δ l ) -φ j (X(t)) .
Note that when i ̸ = j,

(φ i (x -δ k + δ l ) -φ i (x)) (φ j (x -δ k + δ l ) -φ j (x)) is negative if {k, l} =
{i, j} and is zero otherwise. Hence ⟨M i , M j ⟩ t ≤ 0, ∀i ̸ = j. Consequently, for all t positive,

⟨M, M ⟩ t ≤ 1 n 2 n i=1 ⟨M i , M i ⟩ t .
Let α 2 be defined as in Lemma 18, and let U be the exit time from {∥•∥ ∞ ≤ α 2 log n}. Note that, almost surely, for any u ≥ 0, ∆M (u) = 0 or there exist k, l

∈ [n] such that ∆M (u) = φ(X(u) -δ k + δ l ) -φ(X(u)) = 1 n (e θ 2 -1)(e θ 2 X l (u-) -e θ 2 (X k (u-)-1) ).
In either case, before time U , almost surely,

|∆M (u)| ≤ 1 n (e θ 2 -1)e θ 2 ∥X(u-)∥ ∞ ≤ (e θ 2 -1)n θ 2 α 2 -1 .
Similarly,

|φ i (x -δ k + δ l ) -φ i (x)| ≤ (e θ 2 -1)e θ 2 ∥x∥ ∞ (1 {k=i} ∨ 1 {l=i} ).
Hence for any u < U ,

|φ i (x -δ k + δ l ) -φ i (x)| 2 ≤ (e θ 2 -1) 2 n 2θ 2 α 2 (1 {k=i} ∨ 1 {l=i} ).
Then by dividing the double sum to the sum where k = i or l = i or k ̸ = i ̸ = l, we get

M U i , M U i t = t∧U 0 1≤k,l≤n r(X k (u)) n (φ i (X(u) -δ k + δ l ) -φ i (X(u))) 2 du ≤ t∧U 0 (r(X i (u)) + ζ(u))(e θ 2 -1) 2 n 2θ 2 α 2 du,
where we recall that ζ(u) is the mean-field jump rate. Taking the sum over i ∈ [n], we get

M U , M U t ≤ 1 n 2 t∧U 0 2nζ(u) n 2θ 2 α 2 (e θ 2 -1) 2 du ≤ 1 n 2 t∧U 0 2nκ n 2θ 2 α 2 (e θ 2 -1) 2 du = O tn 2θ 2 α 2 -1 ,
where we have used (18) in the second inequality. Now for θ 2 small enough, 2θ

2 α 2 -1 < -1/2, which implies ∆M (u) = O (1/ √ n) and M U , M U t = O (1/ √ n) if u < U and t = (log n) 2 .
Then we apply Lemma 5 to the martingale

M U ∧(log n) 2 (•) := M (U ∧ (log n) 2 ∧ •), with a = 1, K = b 2 = O (1/ √ n), to obtain P x sup s≥0 M U ∧(log n) 2 (s) ≥ 1 ≤ e -Ω( √ n) .
Using the same argument for -M , and taking a union bound, we deduce that

P x sup s≥0 M U ∧(log n) 2 (s) ≥ 1 ≤ 2e -Ω( √ n) .
Taking a union bound with the event in Lemma 18, we get

P x ∃s ∈ [0, (log n) 2 ] : |M s | ≥ 1 ≤ P x sup s≥0 M U ∧(log n) 2 (s) ≥ 1 + P x U ≤ (log n) 2 = O n -3 ,
which finishes our proof.

Path coupling via tagged particles

For k ∈ Z + , define ∆(k) := r(k + 1) -r(k) ≥ 0. (41)
Let Θ be a Poisson point process of intensity

1 n dt ⊗ du ⊗ Card on R + × R + × [n]
, independent of the Poisson processes used in the graphical construction of X. For a site i ∈ [n], define an [n]-valued process I = (I(t)) t≥0 by setting I(0) = i and for each (t, u, k) in Θ,

I(t) := k if u ≤ ∆(X I (t-)) I(t-) otherwise, (42)
where X I (t) := X I(t) (t). This definition means that conditionally on X, the process I(t) will jump with the time-varying rate ∆(X I (t)), and the destination is uniformly chosen among all sites. A simple but important observation is that (X(t) + δ I(t) ) t≥0 has the same distribution as a Zero-Range process starting at x + δ i . We call I a tagged particle, and we stress here that the construction of I relies strictly on condition [START_REF] Lacoin | Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion[END_REF]. For j another site, similarly we can construct a second tagged particle J starting from J(0) = j using the same process Θ. Thus we have a coupling (X(t) + δ I(t) , X(t) + δ J(t) ) t≥0 of two Zero-Range processes starting at x + δ i and x + δ j respectively.We call the particles of X non tagged particles. We note P x,i,j for the law of the process (X, I, J) starting from (x, i, j) and E x,i,j for the expectation taken w.r.t P x,i,j . Let τ be the coalescence time of I and J: τ := inf{t ≥ 0 :

I(t) = J(t)}. (43)
By the classical relation between d tv (•, •) and coupling, we have:

d tv P t x+δ i , P t x+δ j ≤ P x,i,j [τ > t] . (44)
By construction, if the two tagged particles manage to jump at the same time, then coalescence occurs. However, the jump rates of the tagged particles depend on their number of cooccupants, which complicates our task. We will need to analyze carefully the trajectories of (X, I, J) to obtain a good estimate on τ . The inequality (44) gives us the estimate on the total variation distance of two processes starting from two adjacent configurations. However, comparing processes starting from two arbitrary configurations directly by coupling is in general not easy. Nevertheless, if we are interested only in total variation, and if the comparison using coupling of processes from adjacent configurations is simpler, we can extend the comparison to two arbitrary configurations by choosing a path between them and then use triangle inequality. This simple but powerful idea, originally due to Bubley and Dyer in [START_REF] Bubley | Path coupling: A technique for proving rapid mixing in markov chains[END_REF], is the strategy that we will implement. Throughout this subsection, α 1 will be as in Proposition 10, and θ 2 , β 2 , L 2 will be as in Proposition 19. Proposition 19 and (34) imply that starting from any configuration x such that ∥x∥ ∞ ≤ α 1 log n, the system will reach the set {φ θ 2 ≤ L 2 } quickly (in time O (log n)) and then remains in {φ θ 2 ≤ L 2 + 4} for a long time (namely Ω((log n) 2 )). We will prove that if this is the case, then the coalescence time τ is likely to be O (log n).

From now on, let Good := {φ θ 2 ≤ L 2 + 4}. We say that a configuration x is good if x ∈ Good and x is bad otherwise. We introduce the process (X * , I * ) taking values in Good × [n] whose infinitesimal generator L * acts on an observable φ : Good × [n] → R by:

L * φ(x, i) = k,l r(x k ) n (φ(x -δ k + δ l , i) -φ(x, i)) 1 {x-δ k +δ l ∈Good} + n j=1 ∆(x i ) n (φ(x, j) -φ(x, i)).
(45) This definition means that X * is a Zero-Range process constrained to staying good, and I * jumps with a time-varying rate ∆(X * I (t)) := ∆(X * I * (t) (t)) and chooses its destination uniformly among all sites. We can use the same Poisson processes in the graphical construction of (X, I) and (X * , I * ) to obtain a coupling of them, and we can construct the second tagged particle J * analogously. By their construction, if the processes (X, I, J) and (X * , I * , J * ) start from the same configuration (x, i, j), then they coincide up to the time when X turns bad. For any θ > 0, we define the observable

φ θ * : Good × [n] → R + by φ θ * (x, i) := φ θ i (x) = e θx i .
Lemma 21 (Exponential moment of the number of cooccupants). There exist dimension-free constants c 2 and K such that for any (x, i, j)

∈ Good × [n] × [n], when t ≥ c 2 (x i ∨ x j ), E x,i,j e θ 2 (X * I (t)∨(X * J (t)) ≤ K. ( 46
)
Proof. We will prove that there exist positive dimension-free constants a 2 , b 2 such that It remains to prove (47). It is similar to Lemma 15 except now we have an extra term corresponding to the jump of the tagged particle I, which is controlled by the fact that the system is constrained to staying good. More precisely,

L * φ θ 2 * (x, i) ≤ -a 2 φ θ 2 * (x, i) + b 2 , (47)
L * φ θ 2 * (x, i) = k̸ =i r(x k ) n e θ 2 -1 φ θ 2 * (x, i)1 {x-δ k +δ i ∈Good} + k̸ =i r(x i ) n e -θ 2 -1 φ θ 2 * (x, i)1 {x-δ i +δ k ∈Good} + n j=1 ∆(x i ) n (φ θ 2 * (x, j) -φ θ 2 * (x, i)).
We will bound the three terms above to obtain an upper bound on L * φ θ 2 * : The first term: since 1 n n k=1 r(x k ) ≤ κ by (18), hence,

k̸ =i r(x k ) n e θ 2 -1 φ θ 2 * (x, i)1 {x-δ k +δ i ∈Good} ≤ κ(e θ 2 -1)φ θ 2 * (x, i).
The third term is negative when

x i > 1 θ 2 log(L 2 + 4) since x ∈ Good, hence, ∆(x i ) n j=1 e θ 2 x j n -e θ 2 x i < max k≤ 1 θ 2 log(L 2 +4) ∆(k)(L 2 + 4)=:c,
for some dimension-free constant c.

For the second term, we first observe that there are at most n N sites l such that x l > N ρ, for any constant N > 0, thanks to [START_REF] Morris | Spectral gap for the zero range process with constant rate[END_REF]. On the other hand, if x i > N ρ and x l ≤ N ρ, then φ θ 2 (x-δ i +δ l ) ≤ φ θ 2 (x), so if x is good, then so is x -δ i + δ l . Consequently, as e -θ 2 -1 is negative, we have

k̸ =i r(x i ) n (e -θ 2 -1)φ θ 2 * (x, i)1 {x-δ i +δ k ∈Good} ≤ k̸ =i r(x i ) n (e -θ 2 -1)φ θ 2 * (x, i)1 {x-δ i +δ k ∈Good} 1 {x i >N ρ} ≤ k̸ =i r(⌈N ρ⌉) n (e -θ 2 -1)φ θ 2 * (x, i)1 {x i >N ρ} 1 {x k ≤N ρ} ≤r(⌈N ρ⌉)(e -θ 2 -1)φ θ 2 * (x, i)1 {x i >N ρ} (1 -1/N ) =r(⌈N ρ⌉)(e -θ 2 -1)φ θ 2 * (x, i)(1 -1/N ) + r(⌈N ρ⌉)(1 -e -θ 2 )φ θ 2 * (x, i)(1 -1/N )1 {x i ≤N ρ} .
Now we can prove Proposition 11.

Proof of Proposition 11. Let t = α log n, where α is as in Proposition 25. We say that two configurations are adjacent if they differ only by one jump. (54) implies that for any x, y such that x, y are adjacent and ∥x∥ ∞ ∨ ∥y∥ ∞ ≤ α 1 log n, d tv P t x , P t y = O n -3 . Now for x, y arbitrary such that ∥x∥ ∞ ∨ ∥y∥ ∞ ≤ α 1 log n, we can always connect x and y by a path, i.e. a sequence (ω 0 , ω 1 , ..., ω k ) in Ω such that ω 0 = x, ω k = y, and ω l-1 is adjacent to ω l for 1 ≤ l ≤ k. Furthermore, we can pick one of the shortest paths to make sure that k ≤ m and max

1≤l≤k ∥ω l ∥ ∞ ≤ ∥x∥ ∞ ∨ ∥y∥ ∞ ≤ α 1 log n.
Then by triangle inequality,

d tv P t x , P t y ≤ k u=1 d tv P t ω u-1 , P t ωu ≤ mO n -3 = O n -2 ,
where the last equality is due to [START_REF] Morris | Spectral gap for the zero range process with constant rate[END_REF]. By stationarity of π and convexity of d tv (•, •),

d tv P t x , π ≤ y∈Ω π (
≤ π(∥y∥ ∞ > α 1 log n) + π(∥y∥ ∞ ≤ α 1 log n)O n -2 .
Moreover, by letting t → ∞ in Proposition 10, we obtain π(∥y∥ ∞ > α 1 log n) = O n -2 . Combining it with the above inequality, we deduce the claim.

The Poincaré constant

This section is devoted to proving Theorem 3 and Corollary 4. We first recall a classical lemma for general Markov processes:

Lemma 26 (Lower bound on Poincaré constant). Let Ω be a finite state space, and let L be an irreducible reversible Markov generator on Ω. Fix γ > 0, and suppose that for any (x, y) ∈ Ω × Ω such that L(x, y) > 0, there exists a coupling P x,y of two processes with generator L starting from

x and y such that

E x,y [e γτ ] < ∞,
where τ is the coalescence time of the two processes. Then λ * (L) > γ.

Proof. Let A = max

x,y:L(x,y)>0

d tv P t x , π ≥ γ - 1 t (log A + log |Ω|).
The claim is obtained simply by letting t → ∞.

Thanks to Lemma 26, in order to prove λ * = Ω(1), we just need to prove the following:

Proposition 27 (Exponential moment of coalescence time). There exists a dimension-free constant γ such that for all (x, i, j)

∈ Ω × [n] × [n],
for the coupling of two Zero-Range processes starting from x + δ i , x + δ j using tagged particles as described above,

E x,i,j [e γτ ] < ∞. (56)
Proof. We only need to prove the result for big enough n. Let α 1 be as in Proposition 10. Let θ 1 = 6/α 1 , β 1 be a dimension-free constant, and L 1 = L(θ 1 , β 1 ) as in Lemma 15. Let θ 2 < θ 1 be a constant that satisfies Proposition 19, and let L 2 > L 1 and Good be defined as in subsection 3.2. Let T 1 be the hitting time of the set {φ θ 1 ≤ L 1 }. (34) says that E x e β 1 T 1 is finite for all x ∈ Ω, so we only need to prove the result for x ∈ {φ θ 1 ≤ L 1 }. By abuse of notation, we will note P * (•) (resp. E * (•)) for the maximum of P x,i,j (•) (resp. E x,i,j (•)) taken over all x such that φ θ 1 (x) ≤ L 1 and all (i, j) ∈ [n] × [n]. We will prove that there exists γ such that E * e γ(τ ∧k) is bounded uniformly in k. Then the claim is proved simply by letting k tend to infinity using the Monotone Convergence Theorem.

Let A = Good ∪ {y : ∃x ∈ Good, L(x, y) > 0}. Let T bad be the exit time from Good, and let T 2 := T bad ∧ α 3 log n, where α 3 is defined in Corollary 24. In fact, A is the set of all possible values of X up to time T bad , and in particular, X(T 2 ) ∈ A. By our definitions of the dimension free constants, when n is large enough, {φ Note that T 2 ≤ α 3 log n, and hence E * e γ(τ ∧k) 1 {τ <T 2 } ≤ e γα 3 log n = n γα 3 . We deduce that E * e γ(τ ∧k) ≤ n γα 3 + E * e γ(τ ∧k) 1 {τ ≥T 2 } .

θ 1 ≤ L 1 } ⊂ {φ θ 2 ≤ L 2 } ∩ {∥•∥ ∞ ≤ α 1 log n}.
Conditionally on F T 2 , by the strong Markov property, we have E * e γ(τ ∧k) 1 {τ ≥T 2 } ≤ E * e γ(T 2 ∧k) 1 {τ ≥T 2 } E X(T 2 ) e γ(τ ∧k)

≤ n γα 3 P * [τ ≥ T 2 ] max y∈A E y e γT 1 E * e γ(τ ∧k) ,

where in the last inequality we use the fact that X(T 2 ) ∈ A, almost surely. Putting things together, we obtain E * e γ(τ ∧k) ≤ n γα 3 + O n γα 3 -3 max y∈A E y e γT 1 E * e γ(τ ∧k) .

(57)

Consequently,

E * e γ(τ ∧k) ≤

n γα 3 1 -O (n γα 3 -3 ) max y∈A E y [e γT 1 ]
, provided that the denominator of the right-hand side is positive. Note that for y ∈ A, ∥y∥ ∞ = O (log n), and hence φ θ 1 (y) < n p , for some dimension-free constant p. Then by (34) and Jensen's inequality, for all y ∈ A and γ < β 1 ,

E y e γT 1 ≤ E y e β 1 T 1 γ/β 1 = O n γp/β 1 .
Then the denominator above is 1 -O n γα 3 +γp/β 1 -3 , which is positive when γ is small enough. This finishes our proof.

We now prove Theorem 3 and Corollary 4.

We consider two cases: the case where the density is bounded away from zero :

1 2 ≤ E [X 1 ] = m n ≤ ρ,
and the case of low density:

E [X 1 ] = m n < 1 2 .
In case the density is bounded away from zero, it is easy to deduce from Proposition 8 that E [r(X 1 )] = Θ(1) and Var [X 1 ] = Θ(1), and hence the claim.

In the case of low density, we have Var

[X 1 ] ≥ E [X 1 ] -E [X 1 ] 2 and E [r(X 1 )] ≤ sup k∈Z + r(k) k E [X 1 ].
We deduce that

E [r(X 1 )] Var [X 1 ] ≤ sup k∈Z + r(k) k 1 -E [X 1 ] < 2 sup k∈Z + r(k) k ,
which finishes the proof.

Finally, we compute the Poincaré constant of the transition matrix P in Example 2.

Proof of example 2. Let P 1 be the transition matrix of the simple random walk on Z/pZ. It is not hard to see that both P and P 1 are reversible w.r.t the uniform measures on their domains. Note that, for f 1 , ..., f d some eigenfunctions of P 1 with eigenvalues λ 1 , .

  ) φ(X(t ∧ T )) = φ(X(0)) + t∧T 0 e βu Lφ(X(u)) + βφ(X(u)) du + t∧T 0 e βu dM (u).

  Consequently, by Proposition 19 and Proposition 25,P * [τ ≥ T 2 ] = O n -3 .

Proof of Theorem 3 and

 3 Corollary 4. In[START_REF] Hermon | A version of Aldous' spectral-gap conjecture for the zero range process[END_REF] (more precisely, in Corollary 3 and Lemma 13), the authors prove that for P a doubly stochastic transition matrix on [n],λ * (L) ≤ λ * (L P ) λ * (P ) ≤ (1 -1/n) E [r(X 1 )] Var [X 1 ] ,where the expectation and the variance are taken with respect to the stationary law π of L. Lemma 26 and Proposition 27 readily imply that λ * (L) = Ω(1), and hence so is λ * (L P ) λ * (P ) . It remains to prove that E [r(X 1 )] Var [X 1 ] = O (1) .

  Now for (x, y) ∈ Ω × Ω arbitrary, as L is irreducible, we can connect x and y by a path, i.e. a sequence (ω 0 , ω 1 , ..., ω k ) in Ω such that ω 0 = x, ω k = y, and L(ω l-1 , ω l ) > 0, for 1 ≤ l ≤ k. Picking one of the shortest path ensures that k ≤ |Ω|. Hence by the triangle inequality,

				k
		d tv P t x , P t y ≤	d tv P t ω l-1 , P t ω l ≤ A|Ω|e -γt .
				l=1
	By stationarity of π and convexity of d tv (•, •),
		d tv P t x , π ≤	π(y)d tv P t x , P t y ≤ A|Ω|e -γt .
				y∈Ω
	We deduce that	-	1 t	log max x∈Ω
		E x,y [e γτ ]. Then for any x, y such that L(x, y) > 0,
		d tv P t x , P t

y ≤ P x,y [τ > t] ≤ e -γt E x,y [e γτ ] ≤ Ae -γt .

  .., λ d respectively, the function f : (Z/pZ) d → R defined byf (x 1 , ..., x d ) =generate the space of functions from Z/pZ to R due to reversibility, the functions f of the form above also generate the space of functions from (Z/pZ) d to R. Hence every eigenvalue of P is ofthe form λ = d i=1 λ i d , where λ i , 1 ≤ i ≤ d,are some eigenvalues of P 1 . It is well-known that the eigenvalues of P 1 are cos 2πk p , 0 ≤ k ≤ p (see e.g. Chapter 12 of [9]). Due to reversibility of P , λ * (P ) is the smallest eigenvalue of I -P (see e.g. Chapter 12 of [9]), so it is given by

							d
							f i (x i )
						i=1
	is an eigenfunction of P with eigenvalue λ :=	d i=1 λ i d	. Moreover, as the eigenfunctions of P 1
	λ * (P ) = 1 -	1 d	cos	2π p	+ d -1 =	1 d	1 -cos	2π p	≈	1 d	•	2π 2 p 2 ,
	which finishes our proof.

We sum the three inequalities, and afterwards we take a 2 = -κ(e θ 2 -1) + (1 -1/N )r(⌈N ρ⌉)(1 -e -θ 2 ) and b 2 = c + r(⌈N ρ⌉)e θ 2 N ρ .

We choose N large enough to make a 2 > 0, which is what we needed.

We fix a constant c 2 which satisfies Lemma 21. For any initial configuration (x, i, j) ∈ Good × [n] × [n], we define successively the stopping times (T k ) k≥1 as follows:

Lemma 22 (Bound of τ by T k ). Let (T k ) k≥1 be defined as in (48). Then there exists a dimensionfree constant c 3 such that for any (x, i, j)

Proof. We only need to prove for k = 1, then use induction and the strong Markov property. Let c 2 be the constant used in the definition of (T k ) k≥1 , and let K be the corresponding constant in Lemma 21, and let t = c 2 (x i ∨ x j ) = T 1 -1. By (46) and Chernoff's bound,

for any a > 0. We choose a large enough to make the right-hand side less than 1/2. We will prove that for any (x, i, j) ∈ Good × [n] × [n] such that x i ∨ x j <a, there exists a dimension-free constant c > 0 such that

Assuming for the moment that we have (50), let us prove the lemma. It is not hard to see that

In the second inequality, we have used (50) and the Markov property at time t. We deduce the claim simply by taking c 3 = c/2. It remains to prove (50): Suppose that x is good and x i ∨ x j < a. The scenario is that in a finite time, there is no particle arriving at i and j, and the tagged particles wait for two sites i, j to be completely emptied, and afterwards they jump at the same time. More precisely, we use the mixed graphical construction for the process X as follows: let Ξ and Ψ be two independent Poisson processes defined as in Graphical construction 1 and Graphical construction 2. Consider the process X which starts from x and has the following jumps: for each (t, u, e) ∈ Ψ where e ∈ {i, j},

and for each (t, u, k, l) ∈ Ξ where l ∈ [n] \ {i, j},

Then X is a Markov process with generator L on Ω. Here we use Ψ to indicate the jumps to two sites i, j and Ξ to indicate other jumps. Let

In fact, A is the event that there is no non-tagged particle arriving at two sites i, j up to time 1. If A happens, then B i and B j ensure that all non-tagged particles of two sites i, j jump to [n] \ {i, j} in [0, 1/2]. If A, B i , B j happen, then two sites i, j are empty in [1/2, 1], and event C ensures that the two tagged particles stay at {i, j} in [0, 1/2] then jump at the same time (hence coalescence) in [1/2, 1]. Moreover, the Poisson random variables used in the definitions of these events are independent and have parameters Θ(1). We conclude that the events above are independent and their probabilities are Θ(1). It follows that

which finishes our proof.

Lemma 23 (Exponential moment of T k ). Let c 2 and (T k ) k≥1 be as in (48), and let K be the corresponding constant in Lemma 21. Let θ 3 = θ 2 /c 2 . Then for any (x, i, j)

Proof. By convention, let T 0 = 0. For any k ≥ 2, note that T k-1 is F T k-2 -measurable by its definition. Conditionally on F T k-2 , we have

where the inequality is due to (46). Moreover,

where the last inequality is due to the fact that x ∈ Good. The claim is then obtained by induction.

Corollary 24 (Quick coalescence while staying good). There exists a dimension-free constant α 3 such that for any (x, i, j)

Proof. For any α 3 > 0 and k ∈ Z + , the left-hand side is upper bounded by

for some dimension-free constants K, θ 3 , c 3 , due to Lemma 22, Lemma 23, and Chernoff's bound.

We choose k = O (log n) such that (1-c 3 ) k = O n -3 and α 3 large enough such that (Ke

to get what we wanted.

Now we can finally prove the quick coalescence for a configuration x such that ∥x∥ ∞ ≤ α 1 log n.

Proposition 25 (Quick coalescence). Recall that α 1 is fixed in this subsection. There exists a dimension-free constant α such that for any x such that ∥x∥ ∞ ≤ α 1 log n, for any i, j ∈ [n],

Proof. Let α = α 4 + α 3 , where α 3 is as in Corollary 24, and α 4 is a dimension-free constant that we will choose later. Let T be the hitting time of the set {φ θ 2 ≤ L 2 }. The probability that we want to estimate does not exceed the following sum:

We simply prove that all the terms are O n -3 :

1. The first term: By (34) and Chernoff's bound, it is upper bounded by O n θ 2 α 1 -β 2 α 4 , which is O n -3 when α 4 is large enough.

2. The second term is O n -3 by Proposition 19.

3. The last term: From the time T onward, we couple the processes (X, I, J) and (X * , I * , J * ) starting from (X(T ), I(T ), J(T )) by using the same Poisson processes for their graphical constructions. We observe that up to time (log n) 2 , (X, I, J) and (X * , I * , J * ) coincide. Therefore, this term is O n -3 by the strong Markov property at time T and (53), which finishes our proof.


H T Diep
email: diep@u-cergy.fr

Miron Kaufman

Extended Defects in the Potts-Percolation Model of a Solid: Renormalization Group and Monte Carlo Analysis

Keywords: numbers: 05.10.Ln, 05.10.Cc, 62.20.-x

We extend the model of a 2d solid to include a line of defects. Neighboring atoms on the defect line are connected by springs of different strength and different cohesive energy with respect to the rest of the system. Using the Migdal-Kadanoff renormalization group we show that the elastic energy is an irrelevant field at the bulk critical point. For zero elastic energy this model reduces to the Potts model. By using Monte Carlo simulations of the 3-and 4-state Potts model on a square lattice with a line of defects, we confirm the renormalization-group prediction that for a defect interaction larger than the bulk interaction the order parameter of the defect line changes discontinuously while the defect energy varies continuously as a function of temperature at the bulk critical temperature.

I. INTRODUCTION

Thermodynamics of solids with defects is a topic of current interest. 1,2 In this paper we expand an equilibrium statistical mechanics model 3 of a solid to include extended defects.

Previously we used 4 the realistic anharmonic energy versus atomic distance developed by Ferrante and collaborators 5 in a mean-field type computation. We then evaluated 6 the role of thermal fluctuations by using renormalization-group and Monte-Carlo simulations.

In our model described in Section II, the solid is constituted of harmonic springs. If the energy of such a spring is larger than a threshold, the spring is likely to fail. 7 In the limit of zero elastic energy the model reduces to the Potts model, 8 which has been used to describe correlated and uncorrelated percolation processes. 9,10,11,12 In Section III we present phase diagrams based on the renormalization-group Migdal-Kadanoff scheme. This scheme is of course an approximation for models on regular lattices but as demonstrated by Berker 13 it is exact for hierarchical lattices. This latter feature and its simplicity make the Migdal-Kadanoff scheme quite popular. 14,15,16,17,18 We analyze the stability of fixed points and find that the elastic energy is an irrelevant field in the renormalization-group sense. For this reason, in Section IV we report Monte-Carlo simulations of the model with zero elastic energy, i. e. Potts model on a square lattice with a defect line. Our goal is to verify the renormalization group prediction 19 that for q = 3, 4, on the defect line, at the bulk critical temperature, the order parameter jumps discontinuously from zero (high temperature phase) to a nonzero value (low temperature phase)while the energy varies continuously with temperature. This is interesting since for those q values the bulk transition is continuous. It was argued 19 that this unusual 1d defect transition is due to the infinite range correlations at the bulk critical point. It resembles the Thouless transition 20,21 in a one-dimensional system with inverse squared distance decaying interactions. Our concluding remarks are found in Section V.

II. MODEL

The solid is made of "springs" some of which are live and some are failed upon thermal excitations. All processes are assumed to be reversible, unlike the work of Beale and Srolovitz 22 where springs fail irreversibly. The energy of a "spring" < i, j > is given by the Hooke law:

H ij = -E C + k 2 ( r i -r j ) 2 (1)
where r i is the displacement vector from the equilibrium position of atom i, E C is the cohesive energy, k is elastic constant and a is the equilibrium lattice spacing. If the energy of the spring is larger than the threshold energy E 0 the "spring" is more likely to fail than to be alive. p is the probability that the "spring" is alive and 1 -p the probability that the "spring" breaks. We assume its dependence on energy to be given by the Boltzman weight:

p 1 -p = e - H-E 0 k B T = we -K 2 ( r i -r j ) 2 (2)
where

K = k/k B T and w = e E C +E 0 k B T .
For the extended line of defects the elastic, cohesive and threshold energies may take values different from the rest of the system. Hence while in the bulk the parameters are K and w, on the line of defects they are K d and w d .

We allow for correlations between failing events by using the Potts number of states q, which plays the role of a fugacity controlling the number of clusters. For q = 1 we have random percolation as springs fail independently. The partition function is a sum over all possible configurations of "live" springs:

Z = conf ig q c w B Z config elastic (3)
C is the number of clusters, including single site clusters, and B is number of live "springs".

The restricted partition function associated with the elastic energy for a given configuration of bonds (live "springs") is

Z config elastic = T r r e - H elastic k B T (4)
- H elastic k B T = <i,j> K 2 ( r i -r j ) 2 (5)
In Eq. ( 5) the sum is over all live "springs".

By using the Kasteleyn-Fortuin expansion 23 for Potts model we can rewrite the partition function as

Z = T r σ T r r e -H k B T (6)
The Hamiltonian is

- H k B T = <i,j> [J 1 δ(σ i , σ j )- J 2 2 δ(σ i , σ j )( r i -r j ) 2 ]+ <i,j>def ect [J 1d δ(σ i , σ j )- J 2d 2 δ(σ i , σ j )( r i -r j ) 2 ] (7)
where σ i is a Potts spin taking q values. This mapping is a Gaussian approximation valid when, on the right hand side of Eq. 7, the elastic energy is small compared to the first energy contribution. In our Monte Carlo simulations, we use the Hamiltonian in Eq. ( 7) for integer values of q. The coupling constants J 1 and J 2 are related to the original parameters, w and K, as follows:

J 1 = ln(1 + w) (8)
J 2 = K w w + 1 (9)
J 1d = ln(1 + w d ) (10)
J 2d = K d w d w d + 1 (11)
III.

RENORMALIZATION GROUP

The Migdal-Kadanoff recursion equations 24,25 for two dimensions are obtained by assuming that each atom coordinate varies in the interval (-1/2, 1/2), where the equilibrium lattice constant is 1, and also using the Gaussian approximation (small elastic energy):

w ′ = [1 + U(w, K, q)] 2 -1 (12)
K ′ w ′ = K[1 + U(w, K, q)]U(w, K, q) (13)
w ′ d = [1 + U(w, K, q)][1 + U(w d , K d , q)] -1 (14)
K ′ d w ′ d = 1 2 K d [1 + U(w, K, q)]U(w d , K d , q) + 1 2 K[1 + U(w d , K d , q)]U(w, K, q) (15)
where

U(w, K, q) = w 2 erf ( K/4) q K/π + √ 8w.erf ( K/8) (16)
The above recursion equations represent the Gaussian approximation of the exact solutions for hierarchical lattices. The renormalization group flows in the bulk parameter space (w, K) are governed by the following fixed points at K = 0 (pure Potts model):

i. w = 0 (non-percolating live "springs"),

ii. w = ∞ (percolating network of live "springs"),

iii. w = w c (Potts critical point).

A stability analysis at the bulk Potts critical point, (K = 0, w = w c ) yields the two eigenvalues:

i. the thermal eigenvalue: Λ 1 (for the direction along the K = 0 axis) is always larger than 1, meaning the ww c is a relevant field;

ii. The other eigenvalue associated with the flow along the w = w c line away from the pure model (K =0) is Λ 2 < 1 for all q. This means that there is a line of points in the (w, K) flowing into, and thus is in the same universality class as, the pure Potts critical point (w c , 0).

The bulk phase diagram (Fig. 1), for any given q, in the (w, K) plane shows two phases:

I. solid with mostly live springs, II. "crumbling" solid with failed springs, separated by a critical line in the universality class of the q-state Potts model.

The defect fixed points are obtained from Eqs. ( 14) and ( 15) after setting the bulk fields

w and K at their fixed point values (bulk material is critical): w = w c , K = 0. All the fixed points are obtained for K d = 0. We find fixed points at w d = ∞ and at w d = w c , for all q. For q > q 1 = 3 a third fixed point emerges from the one at w d = ∞ at w d = w * . At q = q 2 = 6.8 (when exponent α = 0) the two finite w d fixed points exchange position.

The stability of the fixed points for perturbations of the defect fields (w ii. the eigenvalue ΛK d < 1 for all fixed points (Fig. 3).

The defect phase diagram (Fig. 4) represented in the plane (w d , K d ) is obtained for the bulk fields at their critical values: w = w c , K = 0.

The plane (w d , K d ) is divided in two regions by a critical line that flows into the fixed Since the elastic energy is irrelevant RG field, we concentrate next on the K = K d = 0 case. The RG analysis 19 of this problem (two-dimensional Potts model with a defect line) predicts for 2 < q ≤ 4 a hybrid phase transition on the defect line: discontinuous order parameter and continuous energy.

IV. MONTE CARLO SIMULATIONS

In this section since we set the elastic couplings to zero in Eq. ( 7), we perform MC simulation of the Hamiltonian

- H k B T = J <i,j> δ(σ i , σ j ) + J d <i,j>def ect δ(σ i , σ j ) ( 17
)
where J is the interaction parameter between the bulk nearest-neighbors (NN) and J d is that between two NN on the defect line (J is equal to J 1 of Eq. 8, and J d is J 1d of Eq. 10).

We consider a square lattice of size N x × N y where N x = 80, 120, 160, 200, ..., 1920 and N y = 40, 60, 80. Each lattice site is occupied by a q-state Potts spin. We place the defect line at the middle of the y side of the sample, i.e. y = N y /2. The length of the defect line is thus N x . The reason why we use a very long length for the defect line stems from the fact that the determination of a transition in a linear chain needs large enough sizes to avoid statistical fluctuations. We use periodic boundary conditions.

Our purpose here is to test the following RG prediction of the previous section: at the bulk transition temperature T c , the defect line undergoes a phase transition where its order parameter is discontinuous but its energy is continuous. The bulk transition temperature is the temperature at which the phase transition of the system without the defect line takes place.

Let us consider the case where q = 4. The critical temperature is given by the exact

formula (k B T c /J) -1 = ln(1 + √ q). With q = 4, one has T c ≃ 0.910239 in units of J/k B .
Note that this value of T c corresponds to the thermodynamic limit, i.e. infinite system size. In MC simulation, we work at finite sizes, so for each size we have to determine the "pseudo" transition which corresponds in general to the maximum of the specific heat or of the susceptibility. The maxima of these quantities need not to be at the same temperature.

Only at the infinite size, they should coincide. The theory of finite-size scaling permits to deduce properties of a system at its thermodynamic limit. We have used in this work a size large enough to reproduce the bulk transition temperature up to the fourth decimal. In order to determine the nature of the phase transition of the defect line, we shall use the technique [START_REF] Ferrenberg | [END_REF] which is known to allow us to distinguish with accuracy the order of the phase transition.

The simulation is carried out as follows. We fix J = 1 hereafter. For each value of J d , using first the standard Metropolis MC method 29 we equilibrate the system of a given size N x × N y at a given temperature T during 10 6 Monte Carlo sweeps (MCS) per spin before averaging physical quantities over the next 2 × 10 6 MCS. We determine the transition temperature at the given size N x × N y by examining the calculated physical quantities such as the internal energy per spin E, the specific heat C v per spin, the Potts order parameter Q and the susceptibility per spin χ. For the bulk q-state Potts model, Q is defined as

Q = q max(Q 1 , Q 2 , ..., Q q ) -1 q -1 (18)
where Q i = n i Nx×Ny (i = 1, ..., q), n i being the number of sites having q i . For the defect line, - (lower curve, line is a guide to the eye) vs temperature T for q = 4, with N x = 1920, N y = 60 and

J d = J = 1.
the order parameter Q d is similarly defined on the defect line, namely

Q d = q max(Q ′ 1 , Q ′ 2 , ..., Q ′ q ) -1 q -1 ( 19
)
where

Q ′ i = n ′ i
Nx (i = 1, ..., q), n ′ i being the number of sites on the defect line having q i . Let us show first in Fig. 5 the energy and the specific heat of the case without defects where the bulk critical temperature is T c ≃ 0.9103 for q = 4 with the size used here (the exact value of T c is 0.910239). The energy of the defect line is shown in Fig.

The order parameters the bulk and of the defect line are shown in Fig. 7 for q = 4, J = 1 and J d = 1.6 and 2.

These figures show that at the "bulk" phase transition temperature T c = 0.9103, while the defect energy is continuous, the order parameter of the defect line undergoes a vertical fall, indicating a discontinuity predicted by the RG analysis shown in the previous section.

For large values of J d , for instance J d = 2, the defect line takes a very long time to become Now, let us examine the case where q = 3. The results are very similar to the case q = 4 shown above. So the conjecture of the RG analysis where the transition of the defect line is of first order for J q > J at T c for q > 3 is verified here. Figure 10 shows the defect order parameter Q d versus T for q = 3 with several values of J d . Note that T c = 0.9949 for q = 3.

Let us show in Fig. 11 the value of Q d taken at T c as a function of J d . As seen, the gap is increased with increasing J d . At this stage, it is worth to mention that due to the usual finite size effect, the order parameters Q and Q d do not vanish above T c : a finite tail exists and decreases with increasing lattice size. For Q, the transition point is, in simulations, taken at the change of curvature of Q, i.e. at the maximum of the corresponding susceptibility. As for Q d , due to its discontinuity at T c , the values shown in Fig. 11 are the upper one at T c . For clarity, the error bars are not shown there, but it is on the second digit, for instance Q d = 0.880 ± 0.020 for J d = 1.6.

We show now the time dependence of the order parameter Q d and the energy. (F) as initial spin state ii) cooling, i.e. using a paramagnetic state (P) as initial condition.

We discuss first the energy case. At T c , heating from the F state (bottom curve in Fig. 12) and cooling from the P state (second curve from the bottom in Fig. 12) give the same energy only after two millions MC steps/spin (see the first million MC steps in Fig. 12).

At T = 1, one needs almost the same MC time to get the same energy for the heating and cooling (curves three and four from the bottom in Fig. 12).

The time dependence of Q d is interesting (Fig. 13):

i) At T c , Q d with F condition stays stable (first curve from top in Fig. 13) and reaches the stationary value 0.88 at two millions MC steps/spin. This means that the defect line is ordered at T c while the bulk spins are disordered. However, when cooled from the P state (third curve from the top), Q d stays very small (disordered state). We conclude that at T c there are two possible values of Q d for the same energy. This explains the gap of Q d at T c shown earlier.

ii) At T = 1, slightly above T c , Q d takes a long time (≃ 500000 MC step/spin) to become disordered in the heating procedure (second curve from the top) while it is disordered all the way in the cooling procedure.

Let us discuss more about the discontinuity of the order parameter Q d at T c . In a standard first-order transition, the ordered and disordered phases coexist at T c yielding a discontinuity in both energy and order parameter. In the case studied here, only Q d is discontinuous, while E d is not. Thus, there is no double-peak energy distribution. To answer the question how Q d can take two values at T c , we have examined the snapshots of the defect line taken during It is obvious that these configurations give the same energy if the number of disordered portions is the same, while Q d is different: the first configuration yields a large value of Q d and the second a zero one. This observation explains why at T c the energy is continuous but the order parameter is not.

The phase transition observed here for the defect line is very interesting in several aspects: i) the nature of the transition is novel in the sense that only the defect order parameter is discontinuous, not the defect energy, nor the bulk order parameter and bulk energy;

ii) note that long-range interaction in one dimension can cause a first-order transition, 35,36,37 while systems with short-range interaction do not show such a phase transition. The present defect line with NN interaction shows thus an exception. We believe that its immersion in a disordered neighboring lines at T c plays a key role in provoking such a pseudo-discontinuous transition.

In this paper we have studied a model of a two-dimensional solid with an extended defect line using the Migdal-Kadanoff renormalization group. Since the elastic energy turns out to be an irrelevant field, we then studied this model for zero elastic energy, i.e. 2d Potts model with a defect line, using Monte Carlo simulations. The renormalization-group analysis of the Potts model on a 2d hierarchical lattice with a defect line, suggests an interesting behavior at the bulk transition T c if the interaction on the defect line J d is larger than the bulk interaction J: the order parameter of the defect line should undergo a discontinuity at T c while the energy is continuous. Our Monte Carlo simulations of the 3-and 4-state Potts model on the square lattice with a defect line confirm the RG prediction.

To conclude, we would like to emphasize two points of interest. First, the existence of a phase transition in a one dimensional system (defect line) is rare. It is induced by the bulk infinite correlations at criticality. Second, the nature of the phase transition is unusual in the sense that the order parameter is discontinuous while the energy is not. This is somewhat similar to the Thouless transition in 1d models with long-range interactions.

M.K. wishes to thank the University of Cergy-Pontoise for hospitality while this work was carried out.

  d , K d ) is determined by i. The eigenvalue Λw d governing flow along K d = 0 axis is larger than unity (i.e. relevant perturbation) at the fixed point K d = 0, w d = max(w c , w * ) and is less than unity (i.e. irrelevant perturbation) at the fixed point K d = 0, w d = min(w c , w * ) (Fig. 2);

FIG. 1 :

 1 FIG. 1: Bulk phase diagram for q = 10 and d = 2, in the plane (w, K).

FIG. 2 :

 2 FIG.2: Renormalization group eigenvalue Λw d vs q: solid line at defect critical fixed point; dotteddashed line at the non-percolating critical fixed point; dashed line at the percolating critical fixed point. Two vertical lines are q 1 = 3 and q 2 = 6.8 (see text for comments).

FIG. 3 :

 3 FIG.3: Renormalization group eigenvalue ΛK d vs q: solid line at defect critical fixed point; dasheddotted line at the non-percolating critical fixed point; dashed line at the percolating critical fixed point. Since these eigenvalues are less than unity, the elastic energy is an irrelevant field at the three fixed points. Two vertical lines are q 1 = 3 and q 2 = 6.8 (see text for comments).

FIG. 4 :

 4 FIG. 4: Phase diagram at w = w c , K = 0 in plane (w d , K d ). The two critical phases (infinite correlation length) are separated by a defect critical line. In the small w d phase there is zero probability for percolation on the defect line, while in the large w d phase there is a finite probability for percolation on the defect line.

FIG. 6 :FIG. 7 :FIG. 8 :FIG. 9 :

 6789 FIG. 6: Energy per spin E d of the defect line vs temperature T for q = 4, J d = 1.6 with N x = 1920, N y = 60 and J = 1. Note that E d is continuous at T c . See text for comments.

FIG. 10 :FIG. 11 : 3 (

 10113 FIG. 10: Order parameters Q d of the defect line for J d = 2 (upper curve with black circles), J d = 1.4 (middle curve with void circles) and bulk order parameter Q (lower curve with black squares) vs temperature T for q = 3 with N x = 1920, N y = 60 and J = 1. Note that Q d is discontinuous at T c . See text for comments.

Figures 12 andFIG. 12 :

 1212 FIG.12: Time dependence of the energy par spin of the defect line for q = 3 with J d = 2 at T c = 0.9949 and at T = 1. From the bottom: heating to T c , cooling to T c , heating to T = 1, cooling to T = 1. See text for comments.

FIG. 13 :

 13 FIG. 13: Time dependence of the order parameters Q d of the defect line for q = 3 with J d = 2 at T c = 0.9949 and at T = 1. From the top: heating to T c , heating to T = 1, cooling to T c , cooling to T = 1. See text for comments.