1Introduction FrancescaCollet PaoloDaiPra Theroleofdisorderinthedynamicsofcriticalfluctuationsofmeanfieldmodels

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.17(2012), no. 26, 1–40.

ISSN:1083-6489 DOI:10.1214/EJP.v17-1896

The role of disorder in the dynamics of critical fluctuations of mean field models

Francesca Collet

Paolo Dai Pra

Abstract

The purpose of this paper is to analyze how disorder affects the dynamics of critical fluctuations for two different types of interacting particle system: the Curie-Weiss and Kuramoto model. The models under consideration are a collection of spins and rotators respectively. They both are subject to a mean field interaction and embed- ded in a site-dependent, i.i.d. random environment. As the number of particles goes to infinity their limiting dynamics become deterministic and exhibit phase transition.

The main result concerns the fluctuations around this deterministic limit at the crit- ical point in the thermodynamic limit. From a qualitative point of view, it indicates that when disorder is added spin and rotator systems belong to two different classes of universality, which is not the case for the homogeneous models (i.e., without dis- order).

Keywords:Disordered models, Interacting particle systems, Mean field interaction, Perturba- tion theory.

AMS MSC 2010:60K35; 82C44.

Submitted to EJP on November 4, 2011, final version accepted on March 10, 2012.

1 Introduction

Interacting particle systems with mean fieldinteraction are characterized by the complete absence of geometry in the space of configurations, in the sense that the strength of the interaction between particles is independent of their mutual position.

The advantage of dealing with this kind of models is that they usually are analytically tractable and it is rather simple derive their macroscopic equations. Even if the mean field hypothesis may seem too simplistic to describe physical systems, where geometry and short-range interactions are involved, mean field models have been recently ap- plied to social sciences and finance, as in [3, 8, 9, 11, 14, 16].

We briefly introduce the general framework and some of its peculiar features. Bymean fieldstochastic process we mean a familyx^(N)= (x^(N)(t))_t≥0with the following charac- teristics:

∗Dep. de Ciencia e Ingeniería de Materiales e Ingeniería Química, Uni. Carlos III de Madrid, Spain.

E-mail:fcollet@ing.uc3m.es

†Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Italy.

E-mail:daipra@math.unipd.it

• x^(N)(t) =

x^(N₁ ⁾(t), x^(N)₂ (t), . . . , x^(N_N ⁾(t)

is a Markov process withN components, taking values on a given measurable space(E,E);

• Consider theempirical measure

ρN(t) := 1 N

N

X

k=1

δ_x(N) k (t),

which is a random probability on (E,E). Then (ρ_N(t))_t≥0 is a measure-valued Markovprocess.

Although this is by no means astandarddefinition of mean field model, it captures the basic features of the specific models we will consider.

Let (F,F) be a topological vector space, and h : E → F be a measurable function.

Objects of the form

Z

hdρ_N(t) = 1 N

N

X

k=1

h

x^(N_k ⁾(t) are calledempirical averages. In the case the flow(R

hdρ_N(t))_t≥0 is a Markov process, we say R

hdρN(t)is an order parameter. Note that the empirical measure itself is an order parameter (takingF =set of signed measures onE, andh(x) =δx). Whenever possible, it is interesting to find finite dimensional order parameters, i.e. order param- eters for whichF is finite dimensional.

One of the nice aspects of mean field models is that, in many interesting cases, one can prove aLaw of Large Numbers(asN →+∞) for the order parameters, and character- ize the deterministic limit as a solution of an ordinary differential equation. This limit is often called theMcKean-Vlasovlimit. In particular, the differential equation describing the limit evolution of the empirical measure, will be referred to as theMcKean-Vlasov equation. This equation has the form

d

dtq=Lq,

whereL is a nonlinear operator acting on signed measures onE (even though other spaces may be more convenient for the analysis ofL).

Our main interest is the study of the fluctuations of the order parameter around its lim- iting dynamics. We can capture different features of these fluctuations depending on whether or not the time is rescaled withN. If time is not rescaled and we consider the evolution in a time interval[0, T], withT fixed, a Central Limit Theorem holds for the order parameter for all regimes; in other words, the fluctuations of the order param- eter converge to a Gaussian process, which is the unique solution of a linear diffusion equation. Whenever time is rescaled in such a wayT goes to infinity asN does, we may observe different behaviors. To avoid further complications, we assume the Markov process x^(N)(t) has a “nice” chaotic initial condition: x^(N₁ ⁾(0), x^(N)₂ (0), . . . , x^(N_N ⁾(0) are i.i.d. with common law q0(dx), where q0 is a stationary, locally stable solution of the McKean-Vlasov equation (the system is inlocalequilibrium).

• Subcritical regime. Supposeq0 is theuniquestationary solution of the McKean- Vlasov equation, and it islinearly stable(i.e. stable for the linearized equation).

Then we expect the Central Limit Theorem holdsuniformly in time; in particular, this provides a Central Limit Theorem for the stationary distribution ofx^(N). Some results in this direction are shown in [13].

• Supercritical regime. Suppose the set of stationary, linearly stable solutions of the McKean-Vlasov equation has cardinality greater than1. In this casemetastability phenomena occur at a time scale exponentially growing inN.

• Critical regime. This is the case in the boundary of the subcritical regime: denot- ing byLthe linearization ofLaroundq₀, the spectrumSpec(L)ofLis contained in {z ∈ C : Re(z) ≤ 0}, but there are elements on Spec(L) with zero real part.

Under a suitable time speed-up, the elements of the corresponding eigenspaces may exhibit large and, possibly, non-normal fluctuations (see [10, 6]).

Of course the three regimes described above do not cover in general all possibilities, since stable periodic orbits or even stranger attractors may arise. Moreover, the same model could be in different regimes depending on the values of some parameters (phase transition).

The main subject of this paper is the analysis of the dynamics of the critical fluctua- tions in disordered mean field models.

We consider a mean field model and we add a site-dependent, i.i.d. random environ- ment, acting as an inhomogeneity in the structure of the system; we aim at analyzing the effect of the disorder in the dynamics of critical fluctuations, as compared with the homogeneous case. We deal with the Curie-Weiss and the Kuramoto models. We are not aware of similar results concerning non-equilibrium critical fluctuations in presence of disorder. Static fluctuations for the random Curie-Weiss model have been studied in [1].

We now give the basic ideas of how the dynamics of critical fluctuations are de- termined. As we mentioned above, the deterministic limiting dynamics of the order parameter is described by a nonlinear evolution operatorL. The linearization of this equation around a stationary solution gives rise to the so called linearized operatorL. This operator is also related to the normal fluctuation of the process. At the critical point this operator has an eigenvalue with zero real part, while all other elements of the spectrum have negative real part. The eigenspace of the eigenvalue with zero real part will be calledcritical direction, and usually happens to have low dimension: critical phenomena involve the empirical averages corresponding to this subspace. Thus, our analysis follow the following points.

• Locating the critical direction.

• Determining the correct space-time scaling for the critical fluctuations. This re- quires an approximation of the time evolution of the order parameter that goes beyond the normal approximation.

• Proving that the rescaled fluctuations vanish along non-critical directions. This will be done using the method of “collapsing processes” : it was developed by Comets and Eisele in [6] for a geometric long-range interacting spin system and was previously applied to a homogeneous mean field spin-flip system in [20].

• Determining the limiting dynamics in the critical direction. It will be done using arguments of perturbation theory for Markov processes, which has been treated in [19], and of tightness, applied to a suitable martingale problem.

>From a qualitative point of view, our results indicate that when disorder is added, spin systems and rotators belong to two different classes of universality, which is not the case for homogeneous systems. Roughly speaking, in spin systems the fluctuations produced by the disorder always prevail in the critical regime: these fluctuations evolve in a time scale of orderN¹⁴, while the critical slowing down for homogeneous systems is N¹². For rotators, the disorder does not modify theN¹² slowing down. However, as the

“strength” of the disorder increases, the Kuramoto model undergoes a further phase transition: for sufficiently small disorder, the dynamics of critical fluctuations converge to a nonlinear, ergodic diffusion, as in the homogeneous case; for larger disorder, the limiting diffusion loses ergodicity, and actually explodes in finite time.

We finally remark that in [5] we have analyzed the critical fluctuations for a spin system close in spirit to the Curie-Weiss model, although with a less general disorder distribu- tion.

2 The Random Curie-Weiss Model

2.1 Description of the Model

Let S ={−1,+1}be the spin space, andµ be an even probability onR^{. Let also} η= (ηj)^N_j=1∈R^N be a sequence of independent, identically distributed random vari- ables, defined on some probability space(Ω,F, P), and distributed according toµ. They represent a random, inhomogeneous magnetic field.

Given a configurationσ = (σj)^N_j=1 ∈S^N and a realization of the magnetic fieldη, we define the HamiltonianH_N(σ, η) :S^N ×R^N →R^as

HN(σ, η) =− β 2N

N

X

j,k=1

σjσk−β

N

X

j=1

ηjσj, (2.1)

whereσ_j is the spin value at sitej, and η_j is the local magnetic field associated with the same site. Letβ > 0 be the inverse temperature. For givenη, σ(t) = (σ_j(t))^N_j=1, witht≥0, is aN-spin system evolving as a continuous time Markov chain onS^{N, with} infinitesimal generatorLN acting on functionsf :S^N →Ras follows:

L_Nf(σ) =

N

X

j=1

e^{−βσj(mσN+ηj)}∇^σ_jf(σ), (2.2)

where∇^σ_jf(σ) =f(σ^j)−f(σ)and thek-th component ofσ^j, which is the spin flip at the sitej, is

σ^j_k=

σk for k6=j

−σk for k=j .

The quantitye^{−βσj(mσN+ηj)}represents the jump rate of the spins, i.e. the rate at which the transitionσj → −σj occurs for somej. The expressions (2.1) and (2.2) describe a system of mean field ferromagnetically coupled spins, each with its own random mag- netic field and subject to Glauber dynamics. The two terms in the Hamiltonian have different effects: the first one tends to align the spins, while the second one tends to point each of them in the direction of its local field.

Remark 2.1. For every value ofη,(2.2)has a reversible stationary distribution propor- tional toexp[−HN(σ, η)].

For simplicity, the initial conditionσ(0)is such that (σ_j(0), η_j)^N_j=1 are independent and identically distributed with lawλ. Note that, since the marginal law of theη_j’s isµ, λmust be of the form

λ(σ, dη) =q₀(σ, η)µ(dη) (2.3)

withq0(1, η) +q0(−1, η) = 1,µ-almost surely. The quantity (σj(t))_t∈[0,T] represents the time evolution on[0, T] ofj-th spin value; it is the trajectory of the singlej-th spin in time. The space of all these paths isD[0, T], which is the space of the right-continuous, piecewise-constant functions from [0, T] to S^{. We endow} D[0, T] with the Skorohod topology, which provides a metric and a Borelσ-field (see [12] for details).

2.2 Limiting Dynamics

We now describe the dynamics of the process (2.2), in the limit asN → +∞, in a fixed time interval[0, T]. Later, the equilibrium of the limiting dynamics will be studied.

These results are special cases of what shown in [7], so proofs are omitted. More details can also be found in [4].

Let(σ_j[0, T])^N_j=1∈(D[0, T])^N denote a path of the system in the time interval [0, T], withT positive and fixed. If f : S ×R→ R, we are interested in the asymptotic (as N →+∞) behavior ofempirical averages of the form

1 N

N

X

j=1

f(σj(t), ηj) =:

Z

f dρN(t), where(ρN(t))_t∈[0,T] is the flow ofempirical measures

ρN(t) := 1 N

N

X

j=1

δ(σ_j(t),η_j).

We may think ofρ_N := (ρ_N(t))_t∈[0,T]as acadlagfunction taking values inM1(S×R), the space of probability measures onS×Rendowed with the weak convergence topology, and the related Prokhorov metric, that we denote bydP(·,·).

The first result we state concerns the dynamics of the flow of empirical measures.

We need some more notations. For a given q : S ×R → R, we introduce the linear operatorLq, acting onf :S ×R→Ras follows:

Lqf(σ, η) :=∇^σh

e^{−βσ(mq+η)}f(σ, η)i , where

mq :=

Z

[q(1, η)−q(−1, η)]µ(dη).

Givenη ∈ R^N, we denote byP_N^η the distribution on(D[0, T])^N of the Markov process with generator (2.2) and initial distributionλ. We also denote by

P_N dσ[0, T], dη

:=P_N^η (dσ[0, T])µ^⊗N dη the joint law of the process and the field.

Theorem 2.2. The nonlinear McKean-Vlasov equation ∂q_t(σ,η)

∂t = L_qtq_t(σ, η)

q0(σ, η) given in (2.3) (2.4)

admits a unique solution inC¹h

[0, T], L¹(µ)Si

, andq_t(·, η)is probability onS^{, for}µ- almost everyη and everyt > 0. Moreover, for everyε > 0 there existsC(ε)> 0such that

PN sup

t∈[0,T]

d_P(ρ_N(t), q_t)> ε

!

≤e^−C(ε)N

forN sufficiently large, where, by abuse of notations, we identifyq_twith the probability qt(σ, η)µ(dη)onS ×R^.

Thus, equation (2.4) describes the infinite-volume dynamics of the system. The next result gives a characterization of stationary solutions of (2.4). Given a stationary solu- tionq∗ of (2.4), the nonlinear mapF(q) :=Lqqcan be linearized aroundq∗in a suitable

Banach space, and the spectrum of the linearized mapDF(q_∗)can be considered. All details for this example, together with the proof of the next result, can be found in [7], Section 4.1. The solutionq_∗is called linearly stable if all the elements of the spectrum ofDF(q_∗)have negative real part.

Lemma 2.3. Let q∗ : S ×R → R, such that q∗(σ,·) is measurable and q∗(·, η) is a probability onS^{. Then}q_∗is a stationary solution of (2.4), i.e.Lq∗q_∗≡0, if and only if it is of the form

q_∗(σ, η) = e^{βσ(m∗+η)}

2 cosh (β(m∗+η)), (2.5)

wherem_∗satisfies the self-consistency relation m_∗=

Z

[q_∗(1, η)−q_∗(−1, η)]µ(dη). (2.6) Moreover,m_∗ = 0is always a solution of (2.6), and the corresponding stationary solu- tionq_∗is linearly stable if and only if

β

Z µ(dη)

cosh²(βη) < 1. (2.7)

Remark 2.4. The transition between uniqueness and non-uniqueness of the solution of (2.6)in general is not related to the change of stability form∗ = 0. If the distribution µis unimodal onR, the two thresholds coincide: the paramagnetic solution is linearly stable when it is unique and unstable when it is not. In case we chooseµ= ¹₂(δ_η+δ_−η), with η > 0, the phase diagram is more complex: when (2.7) fails, the paramagnetic solution of (2.6)is either unstable, and it coexists with a pair of opposite stable ferro- magnetic solutions, or may recover linear stability, coexisting with a pair of unstable ferromagnetic solutions and a pair of stable ferromagnetic ones (see [7] for details). A more generalµmay give rise to arbitrarily many solutions of (2.6).

2.3 Dynamics of Critical Fluctuations

β

Z µ(dη) cosh²(βη) = 1

The results of this section are concerned with thefluctuation flow ˆ

ρ_N(t) :=√

N[ρ_N(t)−q_t], (2.8)

that takes values on the space of signed measures onS ×R. It is very convenient to assume that the process starts inlocal equilibrium, i.e.q0(σ, η) =q∗(σ, η), whereq∗(σ, η) is a stationary solution of (2.4); it should be not hard to extend all next results to a general initial condition. The proofs of all results stated here will be given in Section 5. We first state results valid for all temperatures; later, Lemma 2.8, Proposition 2.9, Theorems 2.10 and 2.12 are restricted to the critical case.

Functions fromS ×Rare all of the formF(σ, η) =γ(η) +σφ(η). However Z

γ(η)dˆρ_N(t) =√ N



 1 N

N

X

j=1

γ(η_j)− Z

γ(η)µ(dη)





does not change in time, and has a Gaussian limit for everyγ ∈ L²(µ). Thus, we are only interested in the evolution of integrals of the type

Z

σφ(η)dˆρ_N(t).

It is therefore natural to control the action of the generatorL_N on functions ofσandη of the formψ R

σφ(η)dˆρ_N , with

ˆ ρN :=√

N



 1 N

N

X

j=1

δ_(σj,ηj)−q_∗



.

Proposition 2.5. Let ψ : Rⁿ → R be of class C¹, and φ ∈ L²(ν)ⁿ

, where ν is the measure onR^{defined by}

ν(dη) = µ(dη)

cosh(β(m∗+η)). (2.9)

Then L_Nψ

Z

σφ(η)dˆρ_N

= 2

n

X

i=1

∂iψ Z

σφ(η)dˆρN

Z

sinh(β(m_∗+η))φi(η)dˆρN − Z

σLφi(η)dˆρN

+ 2

n

X

i,j=1

∂_ij²ψ Z

σφ(η)dˆρN

Z φi(η)φj(η)

cosh(β(m_∗+η))µ(dη) +o(1), (2.10) where

Lφ_i(η) = cosh(β(m_∗+η))φ_i(η)−β

Z φi(η)

cosh(β(m_∗+η))µ(dη). (2.11) Moreover the remaindero(1)in (2.10) is of the form

R_N Z

H(σ, η)dρˆ_N

(2.12) whereH(σ, η)is the vector-valued function

H(σ, η) = (σφ(η), σ,[cosh(β(m_∗+η))−σsinh(β(m_∗+η))]φ(η),

[σcosh(β(m_∗+η))−sinh(β(m_∗+η))]φ(η)), and

lim

N→+∞ sup

|x|,|y|,|z|,|w|≤M

R_N(x, y, z, w) = 0 (2.13)

for everyM >0.

Proposition 2.5, whose proof consists of a rather standard computation that will be sketched in Section 5, is the essential ingredient for proving a Central Limit Theorem for the empirical flow, i.e. to show that the fluctuation flow converges in law to a Gaus- sian process. The proof of this result requires to identify an appropriate Hilbert space for the fluctuationsρˆ_N (see e.g. [6] for related results). Our main aim is, however, to de- scribe large-time fluctuations at the critical points; the additional technical difficulties arising, have not allowed us to obtained the desired results under the present assump- tions, in particular with no requirements on the field distributionµ(except evenness).

Thus we find it preferable to make the following assumption at this point.

(F) µhasfinitesupportD^.

Under assumption(F), the spaceL²(ν)is finite-dimensional. Together with the follow- ing simple result, this greatly simplifies the analysis of fluctuations.

Lemma 2.6. The operatorLdefined in(2.11)is self-adjoint inL²(ν).

Now, form:=|supp(µ)|, letϕ0, ϕ1, . . . , ϕm−1be a complete set of eigenvectors forL, with eigenvaluesλ0≤λ1≤. . . ≤λm−1. Proposition 2.5, shows that them-dimensional process(R

σϕ_i(η)dˆρ_N(t))^m−1_i=0 is a Markov process, and provides theN →+∞asymptotic of its infinitesimal generator. The classical Corollary 8.7, in Chapter 4 of [12], allows to obtain convergence in law from convergence of the infinitesimal generator, yielding the following Central Limit Theorem, whose standard proof is omitted.

Proposition 2.7. Set X_i^(N)(t) := R

σϕi(η)dˆρN(t). Then, under PN,

X_i^(N)m−1 i=0

con- verges in law to the Gaussian process(X_i)^m−1_i=0 solving the following linear stochastic differential equations

dX_i(t) = [Hi−λ_iX_i(t)]dt+b_idW_i(t) where

•(X0(0), X1(0), . . . , Xm−1(0),H0,H1, . . . ,Hm−1)is a centered Gaussian vector with Cov(Xi(0), Xj(0)) =

Z

ϕi(η)ϕj(η)µ(dη)

− Z

ϕi(η) tanh(β(m∗+η))µ(dη) Z

ϕj(η) tanh(β(m∗+η))µ(dη)

Cov(Hi,Hj) = Z

ϕi(η)ϕj(η) sinh²(β(m∗+η))µ(dη)

− Z

ϕi(η) sinh(β(m∗+η))µ(dη) Z

ϕj(η) sinh(β(m∗+η))µ(dη)

Cov(Hi, Xj(0)) = Z

ϕi(η)ϕj(η) sinh(β(m∗+η)) tanh(β(m∗+η))µ(dη)

− Z

ϕi(η) sinh(β(m∗+η))µ(dη) Z

ϕj(η) tanh(β(m∗+η))µ(dη)

•b²_i :=R

ϕ²_i(η)ν(dη).

•(W_i)^m−1_i=0 are independent standard Brownian motions, that are independent of the vector(X0(0), X1(0), . . . , X_m−1(0),H0,H1, . . . ,Hm−1).

Note that the randomness of the field persists in the limiting dynamics of fluctua- tions, due to the correlated, constant random driftsHi. Observe thatHi≡0ifµ=δ₀, i.e. when the random field is absent.

We now look more closely at fluctuations around the paramagnetic solutionm_∗= 0 at thecritical regime, i.e. for those values ofβfor whichβR

D µ(dη) cosh²(βη)= 1. Lemma 2.8. Assume βR

D µ(dη)

cosh²(βη) = 1 and m_∗ = 0. ThenL is nonnegative, and its kernel is spanned by the function _cosh(βη)¹ .

In the critical regime βR

D µ(dη)

cosh²(βη) = 1, we have λ0 = 0, and λi > 0 fori > 0 (it is actually easily shown that λ_i ≥ 1 for i > 0). It follows that the process X₀(t) in Proposition 2.7 has a variance that diverges ast→ +∞. A sharper description of the large time fluctuations is obtained by considering more “moderate" fluctuations:

˜

ρN :=N⁻¹⁴ρˆN.

The following result improves the expansion given in Proposition 2.5.

Proposition 2.9. Under the same assumptions of Proposition 2.5, and the further con- ditionsβR

D µ(dη)

cosh²(βη) = 1andm_∗= 0, we have L_Nψ

Z

σφ(η)d˜ρ_N

=L⁽¹⁾ψ + 2N⁻¹⁴

n

X

i=1

∂_iψ Z

σφ(η)d˜ρ_N Z

sinh(βη)φ_i(η)dˆρ_N +N⁻¹⁴L⁽²⁾ψ+N⁻¹²L⁽³⁾ψ+o

N⁻¹²

, (2.14) where

L⁽¹⁾ψ:=−2

n

X

i=1

∂iψ Z

σφ(η)d˜ρN

Z

σLφi(η)dρ˜N

L⁽²⁾ψ:=−2β

n

X

i=1

∂iψ Z

σφ(η)dρ˜N

Z σd˜ρN

Z

σsinh(βη)φi(η)d˜ρN

L⁽³⁾ψ:=

n

X

i=1

∂iψ Z

σφ(η)d˜ρN 2β Z

cosh(βη)φi(η)dˆρN

Z σd˜ρN

−β² Z

σd˜ρN

²Z

σcosh(βη)φi(η)d˜ρN +β³ 3

Z φi(η) cosh(βη)µ(dη)

Z σd˜ρN

³#

+ 2

n

X

i,j=1

∂_ij²ψ Z

σφ(η)dˆρ_N

Z φ_i(η)φ_j(η) cosh(βη) µ(dη) Moreover the remainder o

N⁻¹²

in (2.14) is of the form N⁻¹²RN withRN satisfying (2.13).

Note that in Proposition 2.9, functions depending only onη are still integrated with respect toρˆ, rather thanρ˜; indeed, by the standard Central Limit Theorem, those inte- grals with respect toρˆhave a Gaussian limit underPN.

Proposition 2.9 allows to deal easily with the homogeneous caseµ =δ0. Using the notations of Proposition 2.7 we havem = 1,ϕ0 ≡1. Thus, using Proposition 2.9 with n= 1,φ≡1andβ=βc = 1, we easily observe thatL⁽¹⁾ψ=L⁽²⁾ψ≡0, and

L⁽³⁾ψ=−2 3

Z σd˜ρ_N

³ ψ⁰

Z σd˜ρ_N

+ 2ψ⁰⁰

Z σd˜ρ_N

.

Using convergence of generators as in Proposition 2.7 we readily obtain the dynamics of large-time critical fluctuations for the homogeneous model. This result is a simple special case of what obtained in [6].

Theorem 2.10. Assumeµ=δ0, andβ = 1. The stochastic process YN(t) :=

Z

σd˜ρN(√ N t)

converges weakly, underPN, to the unique solution of the stochastic differential equa- tion







dY(t) =−²₃Y³(t)dt+ 2dW(t) Y(0) = 0

whereW is a standard Brownian motion.

As we will see (proofs are in Section 5), the inhomogeneous case requires more sophisticated arguments.

Definition 2.11. We say that a sequence of stochastic processes(ξ_n(t))_n, fort∈[0, T], collapses to zeroif for everyε >0,

n→+∞lim P sup

t∈[0,T]

|ξn(t)|> ε

!

= 0

Theorem 2.12. Assumem_∗= 0,βR

D µ(dη)

cosh²(βη) = 1, and, fori= 0,1, . . . , m−1, let Y_i^(N)(t) :=

Z

σϕi(η)d˜ρN(N¹⁴t), (2.15) whereϕ0, . . . , ϕ_m−1is the basis introduced in Proposition 2.7. UnderPN the processes

Y_i^(N)(t)m−1 i=1

collapse to zero, whileY₀^(N)(t)converges in law to the process Y0(t) := 2Ht,

whereH is a Gaussian random variable, with zero mean and variance Z

D

tanh²(βη)µ(dη).

Thus, the disorder has a dramatic impact on fluctuations at the critical points: fluc- tuations arise at a much shorter time scale (N¹⁴ rather that N¹²), and have the simple form of a linear function with random slope.

3 The Random Kuramoto Model

3.1 Description of the Model

LetI= [0,2π)be the one dimensional torus, andµbe an even probability onR^{. Let} also η = (ηj)^N_j=1 ∈ R^N be a sequence of independent, identically distributed random variables, defined on some probability space(Ω,F, P), and distributed according toµ. Given a configurationx= (xj)^N_j=1∈I^N and a realization of the random environmentη, we can define the HamiltonianH_N(x, η) :I^N ×R^N →R^as

HN(x, η) =− θ 2N

N

X

j,k=1

cos(xk−xj) +ω

N

X

j=1

ηjxj, (3.1)

wherex_j is the position of the rotator at sitejandωη_j, withω >0, can be interpreted as its own frequency. Letθ, positive parameter, be the coupling strength. For givenη, the stochastic processx(t) = (xj(t))^N_j=1, witht≥0, is aN-rotator system evolving as a Markov diffusion process onI^N, with infinitesimal generatorLN acting onC²functions f :I^N →Ras follows:

LNf(x) = 1 2

N

X

j=1

∂²f

∂x²_j(x) +

N

X

j=1

∂HN

∂x_j (x, η)∂f

∂x_j(x)

= 1 2

N

X

j=1

∂²f

∂x²_j(x) +

N

X

j=1

(

ωη_j+ θ N

N

X

k=1

sin(x_k−x_j) ) ∂f

∂xj

(x). (3.2)

Consider the complex quantity

r_Ne^iΨN = 1 N

N

X

j=1

e^ixj, (3.3)

where0≤r_N ≤1measures the phase coherence of the rotators andΨ_N measures the average phase. We can reformulate the expression of the infinitesimal generator (3.2) in terms of (3.3):

LNf(x) = 1 2

N

X

j=1

∂²f

∂x²_j(x) +

N

X

j=1

{ωηj+θrNsin(ΨN −xj)} ∂f

∂x_j(x). (3.4) The expressions (3.1) and (3.4) describe a system of mean field coupled rotators, each with its own frequency and subject to diffusive dynamics. The two terms in the Hamil- tonian have different effects: the first one tends to synchronize the rotators, while the second one tends to make each of them rotate at its own frequency.

For simplicity, the initial condition x(0) is such that (xj(0), ηj)^N_j=1 are independent and identically distributed with lawλ. We assumeλis of the form

λ(dx, dη) =q0(x, η)µ(dη)dx (3.5) withR

Iq0(x, η)dx= 1,µ-almost surely. The quantityxj(t)represents the time evolution on[0, T]ofj-th rotator; it is the trajectory of the singlej-th rotator in time. The space of all these paths isC[0, T], which is the space of the continuous function from[0, T]to I, endowed with the uniform topology.

3.2 Limiting Dynamics

We now describe the dynamics of the process (3.2), in the limit asN → +∞, in a fixed time interval[0, T]. Later, the equilibrium of the limiting dynamics will be studied.

These results are special cases of what shown in [7], so proofs are omitted.

Let (xj[0, T])^N_j=1 ∈(C[0, T])^N denote a path of the system in the time interval [0, T], withT positive and fixed. Iff : I×R → R, we are interested in the asymptotic (as N →+∞) behavior ofempirical averages of the form

1 N

N

X

j=1

f(x_j(t), η_j) =:

Z

f dρ_N(t), where(ρ_N(t))_t∈[0,T] is the flow ofempirical measures

ρN(t) := 1 N

N

X

j=1

δ(x_j(t),η_j).

We may think ofρN := (ρN(t))_t∈[0,T] as a continuous function taking values in M1(I× R), the space of probability measures on I×R endowed with the weak convergence topology, and the related Prokhorov metric, that we denote byd_P(·,·).

The first result we state concerns the dynamics of the flow of empirical measures.

We need some more notations. For a givenq : I×R → R, we introduce the linear operatorLq, acting onf :I×R→Ras follows:

Lqf(x, η) = 1 2

∂²f

∂x²(x, η)− ∂

∂x{[ωη+θrqsin(Ψq−x)]f(x, η)}, (3.6)

where

rqe^iΨq :=

Z

I

Z

e^ixq(x, η)µ(dη)dx.

Given η ∈ R^N, we denote by P_N^η the distribution on (C[0, T])^N of the Markov process with generator (3.2) and initial distributionλ. We also denote by

PN dx[0, T], dη

:=P_N^η (dx[0, T])µ^⊗N dη the joint law of the process and the environment.

Theorem 3.1. The nonlinear McKean-Vlasov equation ∂q_t(x,η)

∂t = Lqtqt(x, η)

q₀(x, η) given in (3.5) (3.7)

admits a unique solution inC¹

[0, T], L¹(dx⊗µ)

, andqt(·, η)is probability onI, forµ- almost everyη and everyt > 0. Moreover, for everyε > 0 there existsC(ε)> 0such that

PN sup

t∈[0,T]

dP(ρN(t), qt)> ε

!

≤e^−C(ε)N

forN sufficiently large, where, by abuse of notations, we identifyq_twith the probability q_t(x, η)µ(dη)dxonI×R^.

Thus, equation (3.7) describes the infinite-volume dynamics of the system. Sinceµis symmetric and the operatorLpreserves evenness, we can suppose the average phase Ψq_t ≡0, without loss of generality. Next result gives a characterization of stationary solutions of (3.7).

Lemma 3.2. Let q_∗ : I ×R → R, such that q_∗(x,·) is measurable and q_∗(·, η) is a probability onI. Thenq∗ is a stationary solution of(3.7), i.e. Lq∗q∗ ≡0, if and only if it is of the form

q_∗(x, η) = (Z_∗)⁻¹·e^{2(ωηx+θr∗cosx)}

e^4πωη Z 2π

0

e^{−2(ωηx+θr∗cosx)}dx +(1−e^4πωη)

Z x 0

e^{−2(ωηy+θr∗cosy)}dy

, (3.8) whereZ∗is a normalizing factor andr∗satisfies the self-consistency relation

r_∗= Z

I

Z

e^ixq_∗(x, η)µ(dη)dx . (3.9) Moreover,r_∗= 0is always a solution of (3.9)and, letting

θ_c =

Z µ(dη)

1 + 4(ωη)² −1

, (3.10)

we have that

1. ifµis unimodal onR, then the solution of (3.7)corresponding tor_∗= 0is linearly stable if and only ifθ < θc;

2. ifµ = ¹₂(δ1+δ₋₁), then the solution of (3.7)corresponding tor_∗ = 0 is linearly stable if and only ifθ < θc∧2.

Remark 3.3. The transitions uniqueness/non-uniqueness of the solution of (3.9) and stability/instability of r_∗ = 0 in general do not occur at the same threshold. It does, however, in the case 1 of the previous Lemma. The phase diagram related to the case 2 is more complicated. We refer to [7] for further details.

Remark 3.4. Ifr_∗= 0the stationary solution(3.8)reduces toq_∗(x, η) := _2π¹ .

Remark 3.5. When synchronized (i.e. corresponding tor∗ 6= 0) stationary solutions exist, establishing their linear stability is technically more challenging than for ther∗= 0case. This issue is dealt with in [2] for the model with no disorder.

3.3 Dynamics of Critical Fluctuations θc =

Z µ(dη)

1 + 4(ωη)² −1!

The results of this section are concerned with thefluctuation flow ˆ

ρ_N(t) :=√

N[ρ_N(t)−q_t], (3.11)

that takes values on the space of signed measures on I×R. It is very convenient to assume that the process starts in the particularlocal equilibriumq0(x, η) =q∗(x, η) =

1

2π, which is the stationary solution of (3.7) corresponding to r_∗ = 0. The proof of the Central Limit Theorem (Proposition 3.7) when q_∗(x, η) is a syncronousstationary solution of (3.7), i.e. withr_∗ 6= 0, should also be not hard, but will not be given here.

The proofs of all results stated here will be given in Section 6.

Ifφis a function fromI×R, we are interested in the evolution of integrals of the type Z

φ(x, η)dˆρ_N(t).

It is therefore natural to control the action of the generatorLN on functions ofxandη of the formψ R

φ(x, η)dˆρN

,with

ˆ ρN :=

√ N



 1 N

N

X

j=1

δ(x_j,η_j)−q∗



=

√ N



 1 N

N

X

j=1

δ(x_j,η_j)− 1 2π



.

Proposition 3.6. Letψ : Rⁿ → Rbe of class C², and φ ∈ (C²([0,2π)× {−1,1}))ⁿ be 2π-periodic in the first argument. Then

L_Nψ Z

φ(x, η)dˆρ_N

=

n

X

i=1

∂_iψ Z

φ(x, η)dˆρ_N Z

Lφ_i(x, η)dρˆ_N + θ

N¹² Z ∂φ_i

∂x(x, η) sin(y−x)dˆρ_Ndρˆ_N

+1 2

n

X

i,k=1

∂²_ikψ Z

φ(x, η)dˆρN

Z ∂φi

∂x(x, η)∂φk

∂x (x, η)dq_∗ + 1

N¹² Z ∂φi

∂x(x, η)∂φk

∂x (x, η)dˆρN

(3.12) where the operator

Lφ(x, η) = 1 2

∂²φ

∂x²(x, η) +ωη∂φ

∂x(x, η) +θ

cosx Z

cosy φ(y, η)dq_∗ + sinx

Z

siny φ(y, η)dq_∗

(3.13) is the linearization ofL, given by (3.6), around the equilibrium distributionq∗.

Unlike the proof of Proposition 2.5, which requires an expansion of the generator, Proposition 3.6 follows by the direct application of the generator; its proof is omitted.

It provides the key computation for the proof of the Central Limit Theorem (Proposition 3.7 below). In order to simplify the analysis, we make the following assumption on the distribution of the random environment.

(H1) µ= ¹₂(δ1+δ₋₁)

Because of the structure of the system, it is reasonable to focus on functions from I×Rof the formsφ(x, η) = cos(hx),sin(hx),ηcos(hx)orηsin(hx), forh≥1integer, and thus on the behavior of

X_h^(1,N)(t) :=

Z

cos(hx)dˆρN(t), X_h^(2,N)(t) :=

Z

sin(hx)dˆρN(t),

X_h^(3,N)(t) :=

Z

ηcos(hx)dˆρ_N(t)andX_h^(4,N)(t) :=

Z

ηsin(hx)dˆρ_N(t).

As for Proposition 2.7, Proposition 3.6 yields the following Central Limit Theorem. Note that the convergence we obtain is under the joint law of process and disorder. A re- markablequenchedversion of this result, i.e. the convergence for afixedvalue of the disorder, has been recently obtained in [17].

Proposition 3.7. Assume(H1) holds. Forr ≥1, consider the following space of se- quences

H_−r=

x=

x⁽¹⁾_h , x⁽²⁾_h , x⁽³⁾_h , x⁽⁴⁾_h

h≥1:kxk_−r<+∞

, where

kxk²_−r:=

+∞

X

h=1

1 (1 +h²)^r

x⁽¹⁾_h

2

+ x⁽²⁾_h

2

+ x⁽³⁾_h

2

+ x⁽⁴⁾_h

2 . UnderPN, onH_−r the process

X_h^(1,N), X_h^(2,N), X_h^(3,N), X_h^(4,N)

h≥1 converges in law to the Gaussian process

X_h⁽¹⁾, X_h⁽²⁾, X_h⁽³⁾, X_h⁽⁴⁾

h≥1solving the following linear stochastic differential equations

dX_h⁽¹⁾(t) = 1

2 θδ1h−h²

X_h⁽¹⁾(t)−hωX_h⁽⁴⁾(t)

dt+ 1

√

2dW_h⁽¹⁾(t) dX_h⁽²⁾(t) =

1

2 θδ1h−h²

X_h⁽²⁾(t) +hωX_h⁽³⁾(t)

dt+ 1

√2dW_h⁽²⁾(t) dX_h⁽³⁾(t) =

−h²

2 X_h⁽³⁾(t)−hωX_h⁽²⁾(t)

dt+ 1

√2dW_h⁽³⁾(t) dX_h⁽⁴⁾(t) =

−h²

2 X_h⁽⁴⁾(t) +hωX_h⁽¹⁾(t)

dt+ 1

√2dW_h⁽⁴⁾(t) whereδ_1his Kronecker delta and

•

X_h⁽¹⁾(0), X_h⁽²⁾(0), X_h⁽³⁾(0), X_h⁽⁴⁾(0)

h≥1is a centered Gaussian vector withCov

X_h⁽ⁱ⁾(0), X_k^(j)(0)

= 0 fori6=jorh6=kandVar

X_h⁽ⁱ⁾(0)

= ¹₂ for anyi, h.

•

W_h⁽¹⁾, W_h⁽²⁾, W_h⁽³⁾, W_h⁽⁴⁾

h≥1are independent standard Brownian motions, that are independent of

X_h⁽¹⁾(0), X_h⁽²⁾(0), X_h⁽³⁾(0), X_h⁽⁴⁾(0)

h≥1.

Note that the randomness of the field appears only through the parameter ω in the dynamics of fluctuations. The only source of stochasticity is due to the Brownian motions.

We now proceed to the analysis of the critical regime, i.e. for θ = θc ∧2, where θc is given in (3.10) and, under (H1), θc = 1 + 4ω². We make the following further assumption.

(H2) ω < ¹₂.

Under assumptions(H1)-(H2), we have sufficient control of the spectrum ofL, as operator in L²([0,2π)× {−1,1}). In particular, L can be diagonalized in the critical regime, as stated in next Lemma.

Lemma 3.8. Under assumptions(H1)-(H2),ker(L)6= 0if and only if θ = 1 + 4ω².In this last case the spectrum ofLis given by

Spec(L) =

0,−1 2 + 2ω²

∪

−k²

2 ±ikω, k∈Z\ {−1,0,+1}

, with corresponding eigenspaces

ker(L) = span

v₁⁽¹⁾, v₁⁽²⁾ Eig −¹₂+ 2ω²

= span

v₁⁽³⁾, v₁⁽⁴⁾ Eig

−^k₂² +ikω

= span

v_k⁽¹⁾, v_k⁽²⁾ Eig

−^k₂² −ikω

= span

v_k⁽³⁾, v_k⁽⁴⁾ , where

v⁽¹⁾₁ (x, η) := cosx−2ωηsinx v₁⁽²⁾(x, η) := sinx+ 2ωηcosx v⁽³⁾₁ (x, η) :=ηcosx+ 2ωsinx v₁⁽⁴⁾(x, η) := 2ωcosx−ηsinx v⁽¹⁾_k (x, η) := sin(kx)−iηcos(kx) v_k⁽²⁾(x, η) := cos(kx) +iηsin(kx) v⁽³⁾_k (x, η) := sin(kx) +iηcos(kx) v_k⁽⁴⁾(x, η) := cos(kx)−iηsin(kx).

(3.14)

In the critical regimeθ=θc= 1 + 4ω²the variance of the processes

U^(1,N)(t) :=X₁^(1,N)(t)−2ωX₁^(4,N)(t)andU^(2,N)(t) :=X₁^(2,N)(t) + 2ωX₁^(3,N)(t), which are the fluctuations of the empirical averages corresponding to the directions generating the kernel of operatorL, diverge ast →+∞.A sharper description of the large time fluctuations is obtained by considering more “moderate” fluctuations:

˜

ρN :=N⁻¹⁴ρˆN.

We will obtain asymptotics, asN →+∞, for the signed measuresρ˜N(√

N t). Note that these measures are completely characterized by their integrals

V_h^(i,N)(t) :=

Z

v_h⁽ⁱ⁾(x, η)d˜ρN(√

N t), (3.15)

withh≥1andi= 1,2,3,4.

Theorem 3.9. Assumeθc= 1+4ω², andω≤ ¹

2√

2. UnderPN the processes

V_h^(i,N)(t)

h≥2, fori= 1,2,3,4,andV₁^(3,N)(t), V₁^(4,N)(t)collapse to zero in the sense of Definition 2.11,

while the process

V₁^(1,N)(t), V₁^(2,N)(t)

converges weakly to the unique solution V⁽¹⁾(t), V⁽²⁾(t) of the stochastic differential equation



















dV⁽¹⁾(t) =−k(ω)V⁽¹⁾(t)h

V⁽¹⁾(t)²

+ V⁽²⁾(t)²i

dt+σ(ω)dW⁽¹⁾(t) dV⁽²⁾(t) =−k(ω)V⁽²⁾(t)h

V⁽¹⁾(t)²

+ V⁽²⁾(t)²i

dt+σ(ω)dW⁽²⁾(t) V⁽¹⁾(0) =V⁽²⁾(0) = 0

where

k(ω) := (1 + 4ω²)²(1−8ω²) 4(1−4ω²)³(1 +ω²), σ²(ω) := 1 + 4ω²

2 ,

andW⁽¹⁾andW⁽²⁾are two independent standard Brownian motions.

In the case ¹

2√

2 < ω < ¹₂ (for whichk(ω)<0), the process V⁽¹⁾(t), V⁽²⁾(t)

explodes in finite time; the convergence above holds for the localizedprocesses: for everyr >0, the process

V₁^(1,N)(t∧TN,r), V₁^(2,N)(t∧TN,r) converges weakly to

V⁽¹⁾(t∧Tr), V⁽²⁾(t∧Tr) , where

TN,r:= inf

t >0 :

V₁^(1,N)(t)2

+

V₁^(2,N)(t)2

≥r

Tr:= inf

t >0 :

V⁽¹⁾(t)² +

V⁽²⁾(t)²

≥r

.

By Theorem 3.9 we can derive the limiting dynamics of the critical fluctuations for the homogeneous modelµ=δ0.They can be obtained as a particular case settingω= 0. Theorem 3.10. Assumeθc= 1.Forh≥1integer, let

Y_h^(1,N)(t) :=

Z

cos(hx)d˜ρ_N(√

N t)andY_h^(2,N)(t) :=

Z

sin(hx)d˜ρ_N(√ N t). Under PN the processes

Y_h^(i,N)(t)

h≥2, for i = 1,2, collapse to zero in the sense of Definition 2.11, while the process

Y₁^(1,N)(t), Y₁^(2,N)(t)

converges weakly to the unique solution of the stochastic differential equation



















dY⁽¹⁾(t) =−¹₄Y⁽¹⁾(t)h

Y⁽¹⁾(t)²

+ Y⁽²⁾(t)²i dt+^√1

2dW⁽¹⁾(t) dY⁽²⁾(t) =−¹₄Y⁽²⁾(t)h

Y⁽¹⁾(t)²

+ Y⁽²⁾(t)²i dt+^√1

2dW⁽²⁾(t) Y₁⁽¹⁾(0) =Y₁⁽²⁾(0) = 0

whereW⁽¹⁾andW⁽²⁾are two independent standard Brownian motions.

4 Collapsing processes

Before giving the details of the proofs of the results stated previously, we briefly present one of the key technical tool: a Lyapunov-like condition, that guarantees a rather strong form of convergence to zero of a sequence of stochastic processes. The first result (Proposition 4.1) we state concerns semimartingales driven by Poisson pro- cesses, whose proof can be found in the Appendix of [6]. In the case where the driving noises are Brownian motions, the result takes a slightly simpler form (Proposition 4.2);

its proof is a simple adaptation of the one in [6], and it is omitted.

Proposition 4.1. Let{ξ_n(t)}_n≥1be a sequence of positive semimartingales on a prob- ability space(Ω,A,P), with

dξn(t) =Sn(t)dt+ Z

Y

fn(t⁻, y)[Λn(dt, dy)−An(t, dy)dt].

Here,Λ_nis a Point Process of intensityA_n(t, dy)dtonR⁺×Y^{, where}Y is a measurable space, andS_n(t)andf_n(t)areAt-adapted processes, if we consider(At)_t≥0a filtration on(Ω,A,P)generated byΛn.

Let d > 1 and Ci constants independent of n and t. Suppose {κn}_n≥1, {αn}_n≥1 and {βn}n≥1, increasing sequences with

κn^d1α⁻¹_n −−−−−→^n→+∞ 0, κ⁻¹_n α_n−−−−−→^n→+∞ 0, κ⁻¹_n β_n−−−−−→^n→+∞ 0 (a1) and

Eh

ξn(0)di

≤C1α^−d_n for alln . (a2) Furthermore, let{τn}n≥1be stopping times such that fort∈[0, τn]andn≥1,

Sn(t)≤ −κnδξn(t) +βnC2+C3 withδ >0, (a3) sup

ω∈Ω,y∈Y,t≤τn

|fn(t, y)| ≤C4α⁻¹_n , (a4) Z

Y

(fn(t, y))²An(t, dy)≤C5. (a5) Then, for anyε >0, there existC₆>0andn₀such that

sup

n≥n0

P sup

0≤t≤T∧τn

ξn(t)> C6

κn^1dα⁻¹_n ∨αnκ⁻¹_n

≤ε . (4.1)

Proposition 4.2. Let{ξ_n(t)}_n≥1be a sequence of positive semimartingales on a prob- ability space(Ω,A,P), with

dξ_n(t) =S_n(t)dt+

mn

X

i=1

f_n(t, i)dW_i(t).

Here,(Wi)^m_i=1ⁿ are independent standard Brownian motions which generate a filtration (At)_t≥0, andS_n(t)andf_n(t, i)areAt-adapted processes.

Let d > 1 and C_i constants independent of n and t. Suppose {κ_n}_n≥1, {α_n}_n≥1 and {βn}_n≥1, increasing sequences with

κ

1

ndα⁻¹_n −−−−−→^n→+∞ 0, κ⁻¹_n αn

−−−−−→n→+∞ 0, κ⁻¹_n βn

−−−−−→n→+∞ 0 (b1) and

Eh

ξ_n(0)^di

≤C₁α^−d_n for alln . (b2)