Infinite system of brownian balls with interaction : the non-reversible case

February 2007, Vol. 11, p. 55–79 www.edpsciences.org/ps

DOI: 10.1051/ps:2007006

INFINITE SYSTEM OF BROWNIAN BALLS WITH INTERACTION:

THE NON-REVERSIBLE CASE^∗

Myriam Fradon

and Sylvie Rœlly

Abstract. We consider an infinite system of hard balls in R^d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.

Mathematics Subject Classification. 60H10, 60K35.

Invited paper accepted September 2005.

Introduction

The aim of this paper is to construct and analyze an infinite system of interacting hard balls undergoing Brownian motions inR^d and starting from a fixed initial condition.

R. Lang [5] constructed in a pioneer paper the reversible solution of an infinite gradient system of Brownian particles (i.e. balls with radius 0, that is reduced to points) submitted to a smooth pair interaction. It is a so-called equilibrium dynamics in statistical physics, since this process has a time-stationary distribution.

J. Fritz solved some years later in [3] the non-reversible case, which occurs when the initial distribution is no more Gibbsian. For this type of systems, the main difficulty comes from a possible explosion (i.e. an infinite number of particles can enter in a finite volume after a finite time).

On another side, a reversible system of infinitely many Brownian hard balls (without external potential) was studied by H. Tanemura [8]. He constructs a unique solution to an infinite-dimensional Skohorod type equation where the hard core situation – balls can not overlap – appears as a local time term in addition to the basic Brownian motion. The (reversible) initial condition is ditributed like a Gibbs measure associated to the hard core potential.

In the present paper, the model is a mixture of both Lang’s and Tanemura’s models. We deal with Brownian motions submitted to the sum of a hard core potential and a smooth finite range pair potential. In [2] we proved

Keywords and phrases. Stochastic differential equation, local time, hard core potential, Gibbs measure, reversible measure.

∗ Ces quelques pages sont d´edi´ees avec reconnaissance et amiti´e `a Nicole, qui nous fit d´ecouvrir les beaut´es de beaucoup de plan`etes math´ematiques, en particulier celle des diffusions infini-dimensionnelles.

1 Laboratoire CNRS 8524, UFR de Math´ematiques, Universit´e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq Cedex, France;[email protected]

2 Institut f¨ur Mathematik, Universit¨at Potsdam, Am Neuen Palais, 14415 Potsdam, Germany;[email protected]

3 On leave of absence Centre de Math´ematiques Appliqu´ees, UMR CNRS 7641, ´Ecole Polytechnique, 91128 Palaiseau C´edex, France.

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2007006

existence and uniqueness of a reversible solution of equation (E), under the condition that the initial distribution is Gibbs with a small mean density of spheres. We propose here the construction of a strong non-reversible solution of (E), in the sense that the initial condition can be any deterministic configuration in a set of allowed configurations which is clearly identified.

Although some techniques in the proof of the main results are similar to those in [2], we adopt a new pathwise approach for the construction of the solution of (E) which is much finer than in [2], where the time-stationarity of the solution was used at several places. Moreover we make explicit in Theorem 1.3 the set of allowed initial configurations, and prove that any Gibbs measure associated with the dynamical interaction carries a.s. this set.

In Section 1 we present the infinite dimensional equation (E) and we state the results. The sequence of approximating solutions is built in Section 2.1. Furthermore, we prove in Section 2 technical estimates needed in Section 3, for the convergence of the approximations. Finally, Section 4 is devoted to complete the proof of the main results.

1. Dynamics and main results

The particles we deal with in the present paper move inR^d, for a fixedd2, endowed with the Euclidian norm denoted by| |. B(y, ρ) will denote the closed ball centered iny∈R^dwith radiusρ≥0 and more generally, for anyA⊂R^d, we define

B(A, ρ) ={y∈R^d such thatd(y, A)ρ}

whered(y, A) denotes the Euclidian distance betweeny andA. The volume of a subsetAinR^dis also denoted by|A|.

The modelization of point configurations may be done in two equivalent ways. The first possibility is to represent an n points configuration in R^d as a subset (with multiplicity) of cardinal n in R^d, that is as an equivalence class on (R^d)ⁿ under the action of the permutation group Σ_n on{1, . . . , n}. The second possibility is to modelize it as a point measure _n

i=1δ_ξi onR^d. More generally, the set of all point configurations inR^d will be the setMof all point Radon measures onR^d:

M=

ξ=

i∈I

δ_ξi such thatI⊂N, ξi ∈R^d and for all Λ compact inR^d, ξ(Λ)<+∞

.

Mis endowed with the topology of vague convergence. By simplicity, we will identify any point measureξ∈ M with the subset ofR^d{ξi, i∈I}corresponding to its support and with the representants of this subset in (R^d)^I, writing for exampleξ_Λ =ξ∩Λ for the restriction of this configuration to Λ⊂R^d, ξηfor the concatenation of both configurationsξandη. M ∩(R^d)ⁿ is the set of allnpoints configurations.

A functiong onMis calledC¹ if for eachγ∈ M,y →g(yγ) =g(δ_y+γ) is aC¹-function onR^d. D_xg(xγ) denotes its derivative aty=x.

We introduce the following notations.

• For Λ⊂R^d,N_Λ is the counting variable onM: N_Λ(ξ) = Card{i∈N, ξ_i ∈Λ}.

• For Λ⊂R^d,BΛ is theσ-algebra onMgenerated by the sets{N_A=n},n∈N,A⊂Λ,Abounded.

• π (resp. π_Λ) is the Poisson process on R^d (resp. on Λ) with intensity measure the Lebesgue measure dy (resp. dy|Λ).

• Forz >0,π^z(resp. π^z_Λ) is the Poisson process onR^d (resp. on Λ) with activityz, that is with intensity measurez dy (resp. z dy|Λ).

The particles we deal with in this paper are not reduced to points but are hard balls or spheres of diameter r, for a fixed r >0. So the set ofallowed configurationsis the following subset ofM:

A={ξ∈ Msuch that∀i=j |ξ_i−ξ_j|r}.

Remark 1.1. We study here the evolution of a particles configuration under the influence of an interaction potential with finite rangeR. Then a fixed particle can interact with at most a finite numberN of particles.

N only depends ondandR/rand is clearly bounded by ^(R+r/2)_(r/2)d^d = (1 + 2R/r)^d. 1.2. Interaction potential and associated Gibbs measures

For a complete description in a general framework of the concepts introduced in this subsection, we refer the reader to [4].

We are dealing with hard balls with diameterrsubmitted to the action of apair potential, which is a function onR^dof class C²with finite rangeR > r,i.e. satisfyingϕ(x) = 0 if|x|Randϕ(x) =ϕ(−x). Due to the hard core situation the values ofϕ(x) may be chosen arbitrarily for|x|< r. In particular, one can assume without restriction that ϕvanishes in a neighborhood from 0 and that∇ϕ(0) = 0. Sinceϕ has compact support, it is bounded from below: the smallest value of interaction between two particles is given by

ϕ= inf

|x|rϕ(x) 0.

If this real constant is zero there exists only repulsion between the balls; if it is negative there exists an attraction domain around each ball.

Theenergyof a configurationξ∈ M submitted to the potentialϕin the compact volume Λ⊂R^d with the boundary conditionη∈ Mis given by:

E_Λ(ξ|η) =

⎧⎪

⎨

⎪⎩ 1 2

ξi,ξj∈Λ

ϕ(ξ_i−ξ_j) +

ξi∈Λ,ηj∈Λ^c

ϕ(ξ_i−η_j) ifξ_Λη_Λc ∈ A

+∞ otherwise

(1)

(the condition ξ_Λη_Λ^c ∈ A corresponds to configurations for which ξ_Λ ∈ A, η_Λ^c ∈ A and no ball of η_Λ^c is overlapping a ball of ξ_Λ). The energy is well defined since both sums contain no more than ^|B(Λ,r/2)|_|B(0,r/2)|N terms, see Remark 1.1. Moreover, e^{−EΛ(ξ|η)} vanishes as soon as the configurationξ_Λη_Λ^c is not allowed.

We now define the setG(z) of Gibbs measures onAassociated to the potential ϕ with activity parameter z ∈ R⁺. For each compact subset Λ ofR^d, let us define a local density function with respect to the Poisson processπ^z_Λ by:

f_Λ^z(ξ|η) = 1

Z_z^Λ,ηexp(−EΛ(ξ|η)) (2)

where the so-called partition function Z_z^Λ,η is the renormalizing constant:

Z_z^Λ,η= e^−z|Λ| 1 + +∞

n=1

zⁿ n!

Λⁿ

exp−EΛ(y₁· · ·y_n|η) dy1· · ·dy_n

.

Due to the hard core, the above series is only a finite sum and 0< Z_z^Λ,η<+∞.

Definition 1.2. A probability measureµonMbelongs to the set G(z) of Gibbs measures on hard balls with activity zand associated potentialϕif and only if, for each compact subset Λ⊂R^d,

dµ(ξ|BΛ^c)(η) =f_Λ^z(ξ|η) dπ^z_Λ(ξ) forµ−a.e.η.

Remark that any Gibbs measure inG(z) has its support included inA. Dobrushin proved in [1], using compact- ness argument, that there exists at least one element inG(z) when the potential contains a hard core component.

Furthermore the set G(z) is convex and compact. About the cardinality of G(z), remarking that the sum of

the hard core and the smooth potentialϕis superstable and lower regular in the sense of Ruelle [6], we do the following remarks:

– Ifz is small enough Ruelle proved that uniqueness holds. In our case, a sufficient condition would be:

z≤e^Nϕ−1(|B(0, r)|+

1Ir<|y|<R|1−e^−ϕ(y)|dy)⁻¹.

– For z large enough it is conjectured (see [4]) – but still not proved – that phase transition occurs:

CardG(z)>1.

1.3. The stochastic equation (E) and statement of the main results

Let (Ω,F, P) be a probability space with a right continuous filtration{Ft}t0 such that eachFt contains allP−negligible sets and let (W_i(t), t0)_i∈Nbe a family of Ft-adapted independent d-dimensional Brownian motions.

Let us denote C(R⁺,M) (respectivelyC(R⁺,A)) the set of continuousM-valued (resp. A-valued) paths on R⁺, endowed with the topology of uniform convergence on each compact time interval.

Letϕbe the smooth pair potential with finite rangeR introduced in the previous subsection. We consider the following infinite gradient system of stochastic equations satisfied by the Brownian balls:

(E)

⎧⎪

⎨

⎪⎩

Fori∈N, t∈R⁺,

X_i(t) =X_i(0) +W_i(t)−1 2

j∈N

_t

0

∇ϕ(Xi(s)−X_j(s))ds+

j∈N

_t

0

(X_i(s)−X_j(s))dL_ij(s)

where

• (X_i(t), t0)_i∈N∈ C(R⁺,A) satisfies|X_i(t)−X_j(t)|rfort0 andi=j;

• (L_ij(t), t0)_i,j∈Nis a family of non-decreasingR⁺-valued continuous processes satisfying:

L_ij(0) = 0, L_ij≡L_ji and L_ij(t) = _t

0

1I_{|Xi(s)−Xj(s)|=r}dL_ij(s), L_ii≡0.

A solution of the system (E) with initial conditionx∈ Ais a family (X_i^x(t), L^x_ij(t), t0, i, j∈N) of processes such that equation (E) is satisfied with X(0) =x.

The main results of this paper are the following theorems.

Theorem 1.3. The stochastic equation (E) admits a solution with values in A for any deterministic initial configuration which belongs to the set A ⊂ A defined by A = {x ∈ A : P(Ω^x₀∩Ω^x₁) = 1} (sets Ω^x₀ and Ω^x₁ are given in (15) and (23)). This solution is unique as element of C ⊂ C(R⁺,A), a subset of paths with some regularity defined in Proposition 4.4.

Theorem 1.4. If the initial configuration of the stochastic equation (E)is random with distributionµ∈ G(z) for some z > 0 and µ(A) = 1, then this solution is time-reversible, that is its law is invariant with respect to the time reversal.

Proposition 1.5. Let z_c be a critical value of the activity given by: z_c = _(Rd^{exp(2N ϕ)}−r^d)|B(0,1)|. Any Gibbs measure µ∈ G(z)with0< z < z_c has its support included inA.

Remark 1.6. The existence of a critical value for the activity z is related to the still open problem of per- colation for the hard core continuous system. The critical valuez_c given here appears for technical reasons in Corollary 2.5, where a percolation type estimate is computed.

r

0 2r 2R

2l

Figure 1. Particles ofη_Λl are represented; the grey area is the domain whereψ^l,η vanishes.

2. Approximating processes and estimates on the paths set

In this whole subsection,l ∈N^∗ is fixed. To simplify we restrict the study of the paths on the time interval [0,1]. It is obvious that all the results in the sequel hold true on any time interval [0, T],T1, up to a change of constants.

2.1. Construction of approximating processes

We construct the approximating processX^l,x in order that it “essentially” stays in the bounded cube Λ_l= [−l, l]^d (in a sense which will be clear soon). To obtain such a behavior, we introduce in the equation (E) a supplementary gradient drift∇ψ^l,ηwhich vanishes in a subset of Λ_land is repulsive outside of Λ_l.

More precisely, for any allowed configuration η ∈ A which support is disjoint to Λ_l, we fix a R⁺-valued functionψ^l,ηonR^d which is C² with bounded derivatives and vanishes on each (and only on)y∈Λ_lsuch that yη is an allowed configuration (see Fig. 1), that is

ψ^l,η(y) = 0 ⇔ y∈Λ_l= [−l, l]^dand yη∈ A ⇔ y∈Λ_l= [−l, l]^d andd(y, η)r.

We extend the definition of ψ^l,η to any configurationη ∈ A by: ψ^l,η =ψ^l,ηΛcl. We also choose the family (ψ^l,η)_lsuch that, for everyη∈ A,

l∈N^∗

R^d1I_ψl,η(y)>0 exp(−ψ^l,η(y)) dy 1. (3)

Forη∈ Aandn∈N^∗, let us now define then-dimensional stochastic differential equation:

(E_n^l,η)

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∀i∈ {1, . . . , n}, for 0≤t≤1, dX_i(t) = dW_i(t)−1

2 j=1,...,n

∇ϕ(X_i(t)−X_j(t)) +

j:ηj∈Λ^c

∇ϕ(X_i(t)−η_j)

dt

−1

2∇ψ^l,η(X_i(t))dt+

j=1,...,n

(X_i(t)−X_j(t))dL_ij(t)

with L_ij ≡ L_ji for all i and j and L_ij(t) = _t

01I_{|Xi(s)−Xj(s)|=r} dL_ij(s). (E_n^l,η) is a n-dimensional stochastic differential equation reflected inA ∩(R^d)ⁿ with gradient drift −¹₂∇β^l,η_n where

β_n^l,η(x₁, . . . , x_n) =

i=1,...,n

ψ^l,η(x_i) +1 2

j=1,...,n j=i

ϕ(x_i−x_j) +

j:ηj∈Λ^c

ϕ(x_i−η_j)

. (4)

Since the drift−¹₂∇β_n^l,η is bounded and Lipschitz continuous, (E_n^l,η) admits a unique strong solution for each initial n-point configurationx∈ A ∩(R^d)ⁿ (see Th. 5.1 of [7]). We denote this solution byX^l,η,n(x,·). For a general initial configurationx∈ A, we extend the above process as follows:

X^l,x(·) =X^l,η,n(x_Λl,·)xΛ^c_l

where η = x_Λ^c

l and n = Card(x∩Λ_l). It is an M-valued (not necessarily A-valued) process with initial configurationx. Particles which are initially in Λ_lmove like the (E_n^l,η)-dynamics and the other ones stay fixed outside Λ_l.

The solution of (E_n^l,η) with initial distribution ν_n^l,η is reversible, whereν_n^l,η is the finite measure defined on (R^d)ⁿ by

dν^l,η_n (x₁, . . . , x_n) = exp(−β^l,η_n (x₁, . . . , x_n)) 1I_A(x₁, . . . , x_n) dx₁. . .dx_n. Q^l,η_n denotes the time-reversible law ofX^l,η,nstarting fromν_n^l,η:

Q^l,η_n =

P(X^l,η,n(x,·)∈.) dν_n^l,η(x).

The probability measureµ^l,η_z on

+∞

n=0

(R^d)ⁿ is defined forA₀×A₁× · · · ×A_n× · · · by:

µ^l,η_z (A₀×A₁× · · · ×A_n× · · ·) = e^−z2dld Z_z^l,η

+∞

n=0

zⁿ

n! ν_n^l,η(A_n) (5)

whereZ_z^l,η=

+∞

n=0

zⁿ

n! ν_n^l,η((R^d)ⁿ) (with the conventionν₀^l,η({∅}) = 1).

Similarly, consider the probability measure on

+∞

n=0

C([0,1],(R^d)ⁿ) defined by

Q^l,η_z = e^−z2dld Z_z^l,η

+∞

n=0

zⁿ n! Q^l,η_n .

This probability measure is time reversal invariant and has a support inA, as a mixing ofA-supported measures.

We now want to prove that the probability of trajectories which interact too much, vanishes asymptotically whenl→+∞. We will use this result to construct the limit of (X^l,x)_lin the Section 3.

Regular paths areω’s such that each particle interacts only with a finite number of particles during a finite time interval ; X^l,x(ω) is then the (unique) solution of a finite dimensional equation. Bad ω’s are paths such that at least one particle interacts with a great number of other ones, either because it moves very fast (see Sect. 2.2), or because it belongs to a large chain of particles where each one interacts with its neighbors (see Sect. 2.3).

2.2. Paths with high velocity

We obtain here an estimate of the probability that a particle moves “too fast”. In order to establish such an estimate, done in Proposition 2.2, we first compute the probability of fast motion between two fixed bounded domains in R^d.

For every bounded subsetsA₀and A₁ ofR^d and everyε >0 andδ∈]0,1], letFm(A₀, A₁, ε, δ) denote the event “at least a particle goes fromA₀to A₁ with an oscillation greater thanεin a time interval smaller than δ”,i.e.

Fm(A0, A₁, ε, δ) ={X ∈ C([0,1],A), ∃is.t. X_i(0)∈A₀, X_i(1)∈A₁ andw(X_i, δ)> ε}

wherew(X_i, δ) = sup

|t−s|<δ 0s,t1

|Xi(t)−X_i(s)|is the usual modulus of continuity of the pathX_i on [0,1].

Lemma 2.1. For each A₀,A₁⊂R^d bounded, eachε >0,δ∈]0,1], η∈ Aandn∈N^∗, we have:

Q^l,η_n (Fm(A0, A₁, ε, δ))) =

P(X^l,η,n(x,·)∈ Fm(A0, A₁, ε, δ)) dν_n^l,η(x) (6) 41ne^−2Nϕ ν^l,η_n−1((R^d)ⁿ⁻¹)1

δ exp

−ε² 5δ _Rd

(1I_A0+1I_A1)e^−ψl,η dy and

Q^l,η_z (Fm(A0, A₁, ε, δ)))

41 z e^{−2N ϕ} 1

δ exp

−ε² 5δ _Rd

(1I_A0+1I_A1)e^−ψl,η dy. (7) From this lemma proved below, we easily deduce an estimate of the probability, under Q^l,η_z , that a particle starting from B(0, K) moves too fast. For everyK ∈N^∗, ε >0 and δ∈]0,1], letFm(K, ε, δ) be the following event:

Fm(K, ε, δ) ={X ∈ C([0,1],A) such that ∃i, Xi(0)∈B(0, K) andw(X_i, δ)> ε}. Proposition 2.2. The following upper bounds hold:

∀K∈N^∗ ∀ε >0 ∀δ∈]0,1] ∀η∈ A

Q^l,η_n (Fm(K, ε, δ)) n C_d e^{−2N ϕ} ν_n−1^l,η ((R^d)ⁿ⁻¹) 1 δ exp

−ε² 6δ

K^d Q^l,η_z (Fm(K, ε, δ)) z C_d e^{−2N ϕ} 1

δ exp

−ε² 6δ

K^d whereC_d is a constant depending only on dimension d. Similarly one has:

∀K∈N^∗ ∀ε >0 ∀δ∈]0,1] ∀η∈ A

Q^l,η_n (Fm(K, ε, δ)) 246ne^{−2N ϕ} ν_n−1^l,η ((R^d)ⁿ⁻¹) 1 δ exp

−ε² 5δ _Rd

e^−ψl,η(y) dy Q^l,η_z (Fm(K, ε, δ)) 246z e^{−2N ϕ} 1

δ exp

−ε² 5δ _Rd

e^−ψl,η(y) dy.

Proof of Lemma 2.1. We first need an estimate ofQ^l,η_n (Fm(A₀, A₁, ε, δ)).

Let (X^l,η,n, L^l,η,n) denote the unique strong solution of (E_n^l,η) starting fromν_n^l,η, and recall that the distri- bution Q^l,η_n ofX^l,η,nis time reversible on [0,1]. By construction, the processes:

W_i(t) = X_i^l,η,n(t)−X_i^l,η,n(0) +1 2

_t

0

∇iβ_n^l,η(X^l,η,n(s))ds

− _t

0

j=1,...,n

(X_i^l,η,n(s)−X_j^l,η,n(s))dL^l,η,n_ij (s),1in,0t1

and W_i(t) = X_i^l,η,n(1−t)−X_i^l,η,n(1) +1 2

₁

1−t∇_iβ_n^l,η(X^l,η,n(s))ds

− ₁

1−t

j=1,...,n

(X_i^l,η,n(s)−X_j^l,η,n(s))dL^l,η,n_ij (s),1in,0t1

are bothn-dimensional Brownian motions starting from 0. Remarking that

∀t∈[0,1] X^l,η,n(t)−X^l,η,n(0) =1 2

W(t) +W(1−t)−W(1)

and using the equality in law between (X^l,η,n(1− ·),W) and (X^l,η,n, W), we obtain:

Q^l,η_n (Fm(A0, A₁, ε, δ)) =

(R^d)ⁿ

P

⎛

⎜⎝

∃ins.t. X_i^l,η,n(x,0)∈A₀, X_i^l,η,n(x,1)∈A₁ and

|t−s|<δsup

0s,t1

|Wi(t)−W_i(s) +W(1−t)−W(1−s)|>2ε

⎞

⎟⎠dν_n^l,η(x)

P(∃ins.t. x_i∈A₀ andw(W_i, δ)> ε) dν_n^l,η(x) +

P(∃ins.t. x_i∈A₁ andw(W_i, δ)> ε)ν_n^l,η(x).

The right hand side is smaller than n

i=1

ν_n^l,η(x_i∈A₀)P(w(W_i, δ)> ε) + n i=1

ν_n^l,η(x_i∈A₁)P(w(W_i, δ)> ε) n P (w(W₁, δ)> ε)

ν_n^l,η(x₁∈A₀) +ν_n^l,η(x₁∈A₁) . We know from Appendix 5 that

P(w(W₁, δ)> ε) 41 δ exp

−ε² 5δ

·

According to the definition (4) ofβ_n^l,η, since a particle interacts with at mostN other particles (cf. Rem. 1.1):

β^l,η_n (x₁, . . . , x_n)ψ^l,η(x₁) + 2N ϕ+β^l,η_n−1(x₂, . . . , x_n) (8) which implies that

ν_n^l,η(x₁∈A₀) =

(R^d)ⁿ

1I_x1∈A0 1I_A(x₁, . . . , x_n) e^{−βl,ηn (x1,...,xn)}dx₁· · ·dx_n e^−2Nϕ ν_n−1^l,η ((R^d)ⁿ⁻¹)

A0

e^−ψl,η(y)dy (9)

and the same result holds forA₁. This leads to the estimate:

Q^l,η_n (Fm(A0, A₁, ε, δ))ne^{−2N ϕ} ν_n−1^l,η ((R^d)ⁿ⁻¹) 41 δ exp

−ε² 5δ _Rd

(1I_A0+ 1I_A1)e^−ψl,η dy;

by summing overnwe obtain the desired result:

Q^l,η_z (Fm(A₀, A₁, ε, δ))) = e^−z2dld Z_z^l,η

+∞

n=1

zⁿ

n! Q^l,η_n (Fm(A₀, A₁, ε, δ))∩(R^d)ⁿ) 41 e^−z2dld

Z_z^l,η z

+∞

n=1

zⁿ⁻¹

(n−1)! ν_n−1^l,η ((R^d)ⁿ⁻¹)

e^−2Nϕ 1 δ exp

−ε²

5δ (1I_A0+ 1I_A1)e^−ψl,η dy

41z e^{−2N ϕ} 1

δ exp

−ε² 5δ _Rd

(1I_A0+ 1I_A1)e^−ψl,η dy.

Proof of Proposition 2.2. Forj in N, leta_j=K+ ε²

δ + 5j. The sequence (a_j)_j increases froma₀=K+√^ε

to +∞. Now forQ=Q^l,η_n orQ=Q^l,η_z consider δ

Q(Fm(K, ε, δ)) = Q(∃i, |X_i(0)|K andw(X_i, δ)> ε)

Q(∃i, |Xi(0)|K andw(X_i, δ)> εand|Xi(1)|a0) +

+∞

j=0

Q(∃i, |Xi(0)|K anda_j<|Xi(1)|aj+1)

But|Xi(0)|K and|Xi(1)|> a_j imply thatw(X_i,1)> a_j−K, so this is smaller than Q(∃i, |Xi(0)|K, |Xi(1)|a0andw(X_i, δ)> ε)

+

+∞

j=0

Q(∃i, |Xi(0)|K, aj<|Xi(1)|< a_j+1 andw(X_i,1)> a_j−K).

Using Lemma 2.1 we obtain:

Q(Fm(K, ε, δ)) C(Q) 1 δ exp

−ε² 5δ

1I_B(0,K)+ 1I_B(0,a0)

e^−ψl,η dy +C(Q)

+∞

j=0

exp

−1 5

ε²

δ + 5j

1I_B(0,K)+ 1I_{B(0,aj+1)B(0,aj)}

e^−ψl,η dy

C(Q)

δ exp

−ε² 5δ

1I_B(0,K)+ 1I_B(0,a0)+

+∞

j=0

e^−j

1I_B(0,K)+ 1I_{B(0,aj+1)B(0,aj)}

e^−ψl,ηdy

withC(Q^l,η_n ) = 41ne^{−2N ϕ} ν_n−1^l,η ((R^d)ⁿ⁻¹) andC(Q^l,η_z ) = 41z e^−2Nϕ. Moreover, forj1:

B(0,aj+1)B(0,aj)

e^−ψl,η(y)dy(a_j+1)^d |B(0,1)|3^d K^d|B(0,1)| max

1,√ε δ

_d

5(j+ 1)^d (10)

and similarly:

1I_B(0,a0)e^−ψl,η(y) dy (a₀)^d |B(0,1)|2^d K^d |B(0,1)| max

1, ε

√δ _d

.

Since

+∞

j=0

e^−j2 and

+∞

j=0

(

j+ 1)^d e^−j2 this leads to:

Q(Fm(K, ε, δ))C(Q) 1 δ exp

−ε² 5δ

|B(0, K)|3×3^dmax

1, ε

√δ _d√

5^d. Finally

Q^l,η_n (Fm(K, ε, δ)) ne^−2Nϕ ν_n−1^l,η ((R^d)ⁿ⁻¹)C_d 1 δ exp

−ε² 6δ

K^d and Q^l,η_z (Fm(K, ε, δ)) ze^{−2N ϕ} C_d 1

δ exp

−ε² 6δ

K^d

whereC_d= 41|B(0,1)|3×3^d √

5^d sup_x∈R+e^−x2/5max(1, x)^de^x2/6.

An alternative bound for Q(Fm(K, ε, δ)) may be obtained using the fact that each indicator function is smaller than one:

Q(Fm(K, ε, δ))C(Q) 1 δ exp

−ε² 5δ _Rd

6 e^−ψl,η(y)dy.

This completes the proof.

2.3. Large chains of interacting particles

Recall that two particles interact instantaneously only if the distance between their centers is smaller than R, the range of the potentialϕ. But more generally, a particle can have an influence on several ones during any small time interval. To modelize this, we introduce the notion of (R+ε)-chain of particles.

Definition 2.3. Letx∈ Aandε >0. Each subset{x1,· · · , x_n}ofxverifying|x1−x₂|R+ε,· · ·,|xn−1− x_n|R+εis called an (R+ε)-chain of particles ofx.

Now, let us fixK ∈N^∗,M ∈N^∗ and ε >0 and let Ch(K, M, R+ε) denote the event that there exists an (R+ε)-chain ofM particles with one end inside ofB(0, K), that is:

Ch(K, M, R+ε) =

x∈ A,∃{x1,· · ·, x_M}subset ofx,|x1|< Kand|x1−x2|R+ε,· · ·,|x_M−1−xM|R+ε . Our aim in this subsection is to estimate theµ^l,η_z -probability that such a chain exists.

Proposition 2.4. For every M ∈N^∗,ε >0,l∈N^∗,η∈ AandK∈R⁺, we have:

µ^l,η_z ^+∞

K=1

Ch(K, M, R+ε)

z |B(0,1)| exp(−2N ϕ) ((R+ε)^d−r^d) _M−1

.

From this proposition, we easily deduce the following corollary used in Section 2.4.

Corollary 2.5. There exists a critical activity z_c = _(Rd^{exp(2N ϕ)}_{−rd)|B(0,1)|} such that for each z > 0, ε ∈]0,1[ and M ∈N^∗,

l∈Nsup^∗ sup

η∈A µ^l,η_z ^+∞

K=1

Ch(K, M, R+ε)

z z_c

(R+ε)^d−r^d R^d−r^d

M−1

. In particular, for anyz < z_c there existsε >0 such that

Mlim→+∞sup

l∈N^∗ sup

η∈A µ^l,η_z ^+∞

K=1

Ch(K, M, R+ε)

= 0 .

Proof of Proposition 2.4. Each configuration in (R^d)ⁿ∩Ch(K, M, R+ε) has exactly _(n−M^n! _)! representants in (R^d)ⁿ such that (x_n−M+1, . . . , x_n) is a fixedM-uple verifying|xn−M+1| < K and |xn−M+1−x_n−M+2|R+ ε,· · ·,|xn−1−x_n|R+ε. Sinceβ^l,η_n (x₁, . . . , x_n) and 1I_A(x₁, . . . , x_n) do not change by permutation of thex_i’s, this leads to:

ν_n^l,η(Ch(K, M, R+ε))

= n!

(n−M)!

(R^d)ⁿ

1I_|xn−M+1|<K 1I_A(x₁, . . . , x_n) e^{−βnl,η(x1,...,xn)}

n−1

i=n−M+1

1I_{|xi−xi+1|R+ε}dx₁· · ·dx_n. Remarking that

1I_A(x₁, . . . , x_n)1I_{|xn−xn−1|r}1I_A(x₁, . . . , x_n−1), using again inequality (8) and integrating with respect tox_n we obtain:

ν_n^l,η(Ch(K, M, R+ε))

n (n−1)!

((n−1)−(M −1))!

(R^d)ⁿ

1I_|xn−M+1|<K

_n−2

i=n−M+1

1I_{r|xi−xi+1|R+ε}

1I_A(x₁, . . . , x_n−1) 1I_{r|xn−xn−1|R+ε}e^{−2N ϕ} e^{−βn−1l,η (x1,...,xn−1)}dx₁· · ·dx_n ne^{−2N ϕ} ((R+ε)^d−r^d)|B(0,1)|ν_n−1^l,η (Ch(K, M−1, R+ε)).

By definition ofµ^l,η_z (see (5))

µ^l,η_z (Ch(K, M, R+ε)) = e^−z2dld Zz^l,η

nM

zⁿ

n! ν_n^l,η(Ch(K, M, R+ε)). (11) Using the above inequality and iterating the result onM, we obtain:

µ^l,η_z (Ch(K, M, R+ε))

e^−2Nϕ ((R+ε)^d−r^d)|B(0,1)| e^−z2dld Z_z^l,η z

nM

zⁿ⁻¹

(n−1)! ν_n−1^l,η (Ch(K, M−1, R+ε) ze^−2Nϕ ((R+ε)^d−r^d)|B(0,1)|µ^l,η_z (Ch(K, M−1, R+ε))

ze^{−2N ϕ} ((R+ε)^d−r^d)|B(0,1)|_M−1 . Since the eventCh(K, M, R+ε) increases as a function ofK

µ^l,η_z ^+∞

K=1

Ch(K, M, R+ε)

z|B(0,1)| exp(−2N ϕ) ((R+ε)^d−r^d)_M−1

.

2.4. Estimates on the set of regular paths

Let B(m, a, ε) denote the following set of bad paths, in which either a particle has a high oscillation in a small time interval or belongs to a large chain of interacting particles:

∀m∈N^∗ ∀a1 ∀ε >0 B(m, a, ε) = ˜B(m, a, ε)∪B˜˜(m, ε) where B(m, a, ε)˜ = X ∈ C([0,1],A), ∃i, w(Xi, 1

m)> ε

4 and∃t1, |Xi(t)|a+ 2m²

!

and B(m, ε)˜˜ =

X ∈ C([0,1],A), ∃k∈ {0, . . . , m−1}, there exists an (R+ε)−chain of particles of X(_m^k)

with diameter greater thanm−R−ε

.

Let us remark that a→B(m, a, ε) is non-decreasing but ˜˜ B(m, a, ε) is not monotone as a function ofmandε.

Our aim in this section is to estimate the probability ofB(m, a, ε) underQ^l,η_z . Proposition 2.6. For each m∈N^∗ anda1:

l∈Nsup^∗ sup

η∈A Q^l,η_z ( ˜B(m, a, ε))z C_d e^{−2N ϕ} a^d m^2d exp

−ε² 96m

where the constant C_d only depends on dimensiond. One also has, form∈N^∗:

l∈Nsup^∗ sup

η∈A Q^l,η_z

B(m, ε)˜˜

m

z |B(0,1)| exp(−2N ϕ) ((R+ε)^d−r^d)[_R+ε^m ]

.

If z < z_c and ε is small enough (depending on z), this implies that the left hand side decreases exponentially fast as a function ofm.

Proof of Proposition 2.6.

Q^l,η_z ( ˜B(m, a, ε)) Q^l,η_z

∃i, w(Xi, 1 m)> ε

4 and|Xi(0)|a+ 3m² +Q^l,η_z

∃i, |X_i(0)|> a+ 3m² and∃t1, |X_i(t)|a+ 2m² .

The second term of the sum is smaller than

+∞

j=1

Q^l,η_z

∃i, a+ (2 +j)m²<|Xi(0)|a+ (3 +j)m² andw(X_i,1)> jm² .