Interacting Particle Systems

and Related Topics


August 27th – 31th, 2012




Talks



Talks will be 45 mn long.


Marton Balázs (Budapest): Second class particles can perform random walks (in some cases).

Authors : M. Balázs, G. Farkas, P. Kovacs, A. Rakos.

Abstract : Two interacting systems of one conserved quantity are known to possess special product shock-measures: the asymmetric simple exclusion process (ASEP) and the totally asymmetric exponential bricklayers process (EBLP). In both models, existing results are:
(1) with the appropriate choice of parameters, the second class particle sees a product shock-measure stationary (Derrida, Lebowitz and Speer for ASEP, B. for EBLP),
(2) such product shock-measures perform ordinary nearest neighbor random walks (sounds strange enough, does it? Belitsky and Schutz for ASEP, B. for EBLP).
Of course the natural question arises: is it the second class particle that performs the nearest neighbor random walks? As this sounds even more strange, decoding this question was the challenging part of the following result: yes, the second class particle, annealed w.r.t. the shock distribution, does perform a nearest neighbor random walk in the above cases. I will explain the result, and show how it includes both (1) and (2) above. The relatively easy proof might also open up the path for proving / disproving similar results in other models.

   [PDF presentation]

Lorenzo Bertini (Roma):


Thierry Bodineau (Paris): Metastability in the dilute Ising model.

Abstract : We will first review the return to equilibrium of the Ising model when a small external field is applied. The relaxation time is extremely long and can be estimated as the time needed to nucleate critical droplets of the stable phase which will invade the whole system. We will then discuss the impact of disorder on this metastable behavior and show that for Ising model with random interactions (dilution of the couplings) the relaxation is faster as the disorder acts as a catalyst.

   [PDF presentation]

Pietro Caputo (Roma): Entropic repulsion and metastability in the SOS model.

Authors : P. Caputo, E.Lubetzky, F.Martinelli, A.Sly, F.L.Toninelli.

Abstract : We discuss the relaxation to equilibrium of a random (2+1)-dimensional SOS interface pinned at the boundary of a lattice box in the low temperature phase. In the presence of a positivity (hard wall) constraint, the interface is pushed away from the wall at a height H(L) proportional to log(L), where L is the side of the box. We give sharp estimates which quantify this phenomenon, known as entropic repulsion. Also, we analyze the effect of entropic repulsion on the dynamics of the interface. We show that relaxation to equilibrium occurs through a sequence of metastable transitions between successive layers until the final height H(L) is achieved. Our analysis shows that the spectral gap is exponentially small in the parameter L. This contrasts with the free case (no wall constraint), where one expects polynomial bounds to hold.

   [PDF presentation]

Francis Comets (Paris): Last Passage Percolation and Traveling Waves.

Authors : F. Comets, J. Quastel, A. Ramirez.

Abstract : We consider a system of N particles with a stochastic dynamics introduced by Brunet and Derrida. The particles can be interpreted as last passage times in directed percolation on {1,...,N} of mean-field type. The particles remain grouped and move like a traveling wave, subject to discretization and driven by a random noise. As N increases, we obtain estimates for the speed of the front and its profile, for different laws of the driving noise. The Gumbel distribution plays a central role for the particle jumps, and we show that the scaling limit is a Levy process in this case. The case of bounded jumps yields a completely different behavior.

   [PDF presentation]

Ivan Corwin (Boston): From duality to determinants.

Authors : A. Borodin, I. Corwin, T. Sasamoto.

Abstract : I will explain duality for ASEP and q-TASEP and then solve the associated PDEs for special initial conditions via a contour integral ansatz. Using the resulting formulas I develop generating series which identify particle location distributions and which can be expressed as Fredholm determinants. Asymptotics of such expressions are readily computable and lead to KPZ universality class fluctuations.

   [PDF presentation]

Bernard Derrida (Paris): Current fluctuations at a phase transition and finite size effects.

Abstract : Non-equilibrium diffusive systems may exhibit phase transitions. One of the models with such a phase transition is the ABC model on a ring, with a broken continuous symmetry. The current fluctuations diverge at the phase transition and the system exhibits anomalous Fourier's law. The talk will present the predictions of the macroscopic fluctuation theory and compare them to the results of numerical simulations. Some preliminary results on finite size corrections to the large deviation function of the density will also be presented.

   [PDF presentation]

Alessandra Faggionato (Roma): Effective dynamics for the East model.

Authors : A. Faggionato, F. Martinelli, C. Roberto, C. Toninelli, P. Chleboun.

Abstract : The East model is a 1D kinetically constrained model, introduced to study glassy dynamics. The main interest is for the regime of small temperature. In this regime the dynamics is subdiffusive and effective dynamics have been proposed. As we recall, when the length of the system is fixed (and therefore does not depend on the temperature), the effective dynamics is given by a coalescence process (as suggested by Evans and Sollich and proved by Faggionato et al.). When the length of the system grows with the inverse temperature, as for example when the system is of the same order of the equilibrium lengthscale, then the analysis becomes extremely intriguing and different effective dynamics have been proposed. The one suggested by Evans and Sollich for equilibrium lenghtscales, called superdomain dynamics, is based on a "continuous timescale separation hypothesis" supported by simulations. We will show that this conjecture is false and discuss possible scenarios.

   [PDF presentation]

Pablo Ferrari (Buenos Aires): Interacting random maps in $\mathbb Z$.

Authors : Joint work in preparation by I. Armendariz, P. Ferrari, P. Groisman, F. Leonardi.

Abstract : Consider a Gibbs measure on the set of bijections $f:\mathbb Z\to\mathbb Z$ with hamiltonian $\sum_x (x-f(x))^2$. The identity is a ground state. We show the existence of infinite volume measures for low temperature by showing that in this case the typical configuration is the identity with perturbations. The perturbations are related to cycle permutations. This model was proposed by Ueltschi and coauthors and worked out in this case by Biskup and Ritchthammer. We use perfect simulation techniques to prove the result.

   [PDF presentation]

Alberto Gandolfi (Firenze): A generalization of the random cluster representation.

Authors : J. van den Berg, A. Gandolfi.

Abstract : The FK random cluster representation has been introduced for the Ising model and later developed for a few other closely related models, becoming a basic tool. Here we introduce a generalization which applies to all Gibbs distributions and beyond.
Such generalization becomes effective when applied to the folding of a distribution, as introduced for product spaces by Reimer in his proof of the BK inequality. The simultaneous use of the general random cluster representation and of the foldings leads to several new results, such as an inequality for four-arm events and the BK property of the antiferromagnetic Curie-Weiss model, and reinterpretations of old ones, such as disagreement percolation or the FKG inequality.

   [PDF presentation]

Alexandre Gaudillière (Marseille): Looking for large cliques through spin glasses.

Abstract : The search problem for the largest cliques in a given graph is an NP-hard problem. Numerical simulations have proven the high efficiency of a recent algorithm for this problem: the cavity algorithm that was introduced by Iovanella, Scoppola and Scoppola. This is a conservative version of a probabilistic cellular automata built on statistical mechanics methods introduced in the study of spin glasses. We will analyze quantitatively the algorithm efficiency for graphs that are generally considered among the more challenging for the largest cliques search problem: Erdös random graphs. We will then have to understand the dynamics of a small cloud of particles in a disordered environment.


Antoine Gerschenfeld (Paris): Anomalous current fluctuations at a phase transition.

Authors : B. Derrida, A. Gerschenfeld.

Abstract : Fourier's law states that the average flow of heat through a system placed between two reservoirs at different temperatures should decrease as the inverse, $1/L$, of its length $L$. Since the 1950s, it has been known numerically that it is violated in some one-dimensional models, in particular when they conserve momentum : the average heat flux across such a system decays instead as a non-integer power of $L$, typically $L^{-2/3}$. More recently, it has been shown that this behavior extends to higher cumulants of the heat flux, as well as to other boundary conditions (such as a closed system on a ring).
Bernard Derrida and I studied the behavior of the current in the ABC model, a 1D, multispecies lattice gas which undergoes a dynamical phase transition between an homogeneous and a modulated phase. We have shown analytically that, while the model obeys Fourier's law in both phases, its conductivity diverges at the transition : around the critical point, current fluctuations become anomalous, with a behavior reminiscent of those of momentum-conserving systems.

   [PDF presentation]

Frank den Hollander (Leiden): A random walk on top of a contact process.

Authors : F. den Hollander, R. dos Santos.

Abstract : We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument within space-time cones that keeps track of the infections generated by single infections. We prove a number of properties of the speed as a function of the infection parameter, and state a number of open problems.

   [PDF presentation]

Milton Jara (Rio de Janeiro): Some generalizations of the KPZ equation.

Authors : C. Bernardin, P. Gonçalves, M. Gubinelli, M. Jara.

Abstract : We will propose some generalizations of the KPZ equation, and we will explain how do they arise as scaling limis of interacting particle systems.

   [PDF presentation]

Claudio Landim (Rouen and Rio de Janeiro): Metastability of the condensate in zero range processes.

Abstract : It is well known that in the stationary state of certain zero range processes all particles but a finite number accumulate on one single site. We show in this talk that in an appropriate time scale the site which concentrates almost all particles evolves as a random walk whose transition rates are proportional to the capacities of the underlying random walk.

   [PDF presentation]

Hubert Lacoin (Paris): The scaling limit of polymer dynamics in the pinned phase.

Abstract : We study a simple heat-bath type dynamic for a simple model of polymer interacting with an interface. The polymer is the graph of a a nearest neighbor path in Z constrained to stay non-negative, and the interaction is modelised by energy penalties/bonuses given when the polymer touches when the path touches 0. This dynamic has been studied by D. Wilson for the case without interaction, then by Caputo et al. for the more general case, both with mixing properties of the system (relaxation time/mixing time). Our work focuses on the scaling limit of the polymer dynamic under diffusive scaling. Whereas in the repulsive phase the interaction with the origin does not appear on the scaling limit (which is the grah of the solution of the heat equation with Dirichlet boundary condition), the presence of an attractive substrate change the picture and the the rescaled picture is the solution of a Stefan free-boundary problem.

   [PDF presentation]

Cyrille Lucas (Nanterre): Internal Diffusion Limited Aggregation, from the centered to the drifted case.

Abstract : Internal Diffusion Limited Aggregation, or iDLA, is a growth model in which random sets are constructed recursively. At each step, a random walk starts at the origin and the first point it visits outside the cluster is added to the aggregate. The asymptotic behavior of this model depends on the properties of the random walk, and limiting shape results are known for a wide range of centered walks. In the case of drifted walks, we will show the existence of an almost sure limiting shape connected to a parabolic PDE.

   [PDF presentation]

Thomas Mountford (Lausanne): A metastability result for the contact processon finite graphs.

Authors : T. Mountford, J.-C. Mourrat, D. Valesin, Q. Yao.

Abstract : We consider the contact process on a graph of bounded degree and show that if the infection parameter exceeds the critical value for one dimension then the extinction time is exponentially large in the number of vertices. We then present an application to the contact process on a NSW graph.


Errico Presutti (Roma): Microstructures in one dimensional systems.

Abstract : Microstructures appear as the outcome of a competition between forces acting on very different spatial scales.
In a continuum mechanics setup the existence of fast oscillations, microstructures, has been established in several models; in statistical mechanics oscillations have been observed studying the ground states of Ising models with a long range repulsive interaction coupled to a short range attractive potential. The extension to positive temperatures is still an open problem.
In my talk I will describe the appearance of microstructures in a one dimensional Ising model with ferromagnetic Kac interactions. Here the competition is between mean field, which, at low temperatures, has a phase transition and entropy which in one dimension always wins against finite range interactions.
Microstructures are found when the magnetic field is very small and are studied by introducing a coarse grained description of the spin configurations, which indicates the local phase of the spins. It is proved that by the introduction of another random variable the coarse grained process becomes a renewal process where the two phases alternate on a suitable spatial scale.

   [PDF presentation]

Fraydoun Rezakhanlou (Berkeley): Symplectic diffusions.

Abstract : Poincaré's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler Equation. Iyer-Constantin Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier-Stokes Equation. Weakly symplectic diffusions are defined to produce stochastically symplectic flows in a systematic away. With the aid of symplectic diffusions, we produce a family of martigales associated with solutions to Navier-Stokes Equation that in turn can be used to prove Iyer-Constantin Circulation Theorem.


Timo Seppäläinen (Madison): Exactly solvable directed polymers in the KPZ universality class.

Abstract : Three exactly solvable 1+1 dimensional directed polymer models are currently known in the KPZ (Kardar-Parisi-Zhang) universality class: the KPZ equation itself, the O'Connell-Yor semidiscrete polymer in a Brownian environment, and the log-gamma lattice polymer. This talk discusses the expected behavior of models in the KPZ class and then some results for the exactly solvable models.

   [PDF presentation]

Sunder Sethuraman (Tucson): A conservative KPZ equation from zero-range and other interactions.

Authors : P. Gonçalves, M. Jara, S. Sethuraman.

Abstract : We derive a type of conservative KPZ equation, in terms of a martingale problem, as a scaling limit of fluctuation fields in weakly asymmetric zero-range and other processes.

   [PDF presentation]

Cristina Toninelli (Paris): Dynamical phase transition for kinetically constrained particle systems.

Abstract : We will introduce a class of interacting particle systems used in physics literature to model glassy dynamics, namely kinetically constrained models (KCM).
A peculiar glassy feature of KCM is the occurrence of an heterogenous dynamics with the coexistence of mobile and blocked regions. In order to characterize this behavior a relevant parameter is the activity which measures the microscopic number of moves per unit time. We will show that the large deviations of the activity exhibit a non-equilibrium phase transition in the thermodynamic limit since reducing the activity is more likely than increasing it due to a blocking mechanism induced by the constraints. We will focus on one dimensional KCM and analyze the finite size effects around this first order phase transition and the phase coexistence between the active and inactive dynamical phases.

   [PDF presentation]

Bálint Tóth (Budapest): Relaxed Sector Condition.

Authors : I. Horvath, B. Tóth, B. Veto.

Abstract : I will present an alternative, weaker formulation of the "sector conditions" of non-reversible Kipnis-Varadhan theory. The "graded sector condition" of Sethuraman-Varadhan-Yau follows naturally, in a less computational way. Some applications will also be shown.

   [PDF presentation]

Daniel Valesin (Vancouver): Metastable Densities for Contact Processes on Random Graphs.

Authors : T. Mountford, D. Valesin, Q. Yao.

Abstract : We consider the contact process on a random graph chosen with a fixed degree, power law distribution, according to a model proposed by Newman, Strogatz and Watts (2001). We follow the work of Chatterjee and Durrett (2009) who showed that for arbitrarily small infection parameter λ > 0 the limiting metastable density does not tend to zero as the graph size becomes large. We show three distinct regimes for this density depending on the tail of the degree law.


Alexander Vandenberg-Rodes (Irvine): Survival and Consensus for the Biased Voter Model.

Authors : O. Louidor, R. Tessler, A. Vandenberg-Rodes.

Abstract : The Biased Voter Model (BVM) is relative of the classical Voter Model in which one opinion is chosen to be more likely to spread than the other. After parameterizing the degree of asymmetry, we consider the survival of an insurgent opinion. In contrast with the process on the $d$-dimensional lattice, the BVM on regular trees shows a rich phase structure similar to the contact process, but with the novel feature of an almost sure limiting consensus.


S.R.S. Varadhan (New York): Quenched Large Deviations and applications.

Abstract : One of the uses of large deviation theory is to prove the existence of and provide a variational formula for limits of the form $$ \lim_{n\to\infty} \frac{1}{n}\log E^P[\exp[\sum_{1=1}^n F_i]] $$ for certain classes of functions $F_i(\omega)$ and certain classes of probability distributions P. In studying quenched large deviations there is an extra variable in that $F=F(\omega_1,\omega_2)$ and the integration is only over $\omega_1$. The limit therefore depends on $\omega_2$. The problem is to examine this dependence. An example is to investigate, for a nice Markov chain $\{X_i\}$, under what conditions on $\{a_i\}$, the limit $$ \frac{1}{n}\log E^P[\exp[\sum_{i=1}^n f(a_i,X_i)] ] $$ exists and what is it? This can then be used to examine limits of the form $$ \frac{1}{n}\log E^P[\exp[\sum_{i=1}^n f(X_i,X_{2i})] ] $$ for nice Markov chains.

   [PDF presentation]

Nikolaos Zygouras (Warwick): Tropical combinatorics and random polymers.

Authors : I. Corwin, N. O'Connell, T. Seppalainen, N. Zygouras.

Abstract : The Robinson-Schensted-Knuth algorithm maps a weight matrix onto a semi-standard Young tableaux. This algorithm can be encoded in terms of the algebra of operators generated by (max,+). We will present a tropical version of the algorithm, encoded by the algebra of operators generated (+,x) and we will show how this tropical version can be used to identify the distribution of the partition function of a directed polymer model with inverse gamma disorder.

   [PDF presentation]


Posters


Oriane Blondel (Paris): Front progression in the East model.

Abstract : The East model is a one-dimensional interacting particle system in which a flip transition is allowed only if the right neighbour is empty. We study this model starting from a configuration with finitely many zeros on the left. We are interested in the behaviour of the left-most zero (the front) and the distribution of the configuration as seen from the front.

   [Poster]

Francesca Collet (Bologna): The role of disorder in the dynamics of critical fluctuations of mean field models.

Authors : F. Collet, P. Dai Pra.

Abstract : We aim at analyzing how the disorder affects the fluctuation dynamics 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 embedded 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 concern the fluctuations around this deterministic limit at the critical 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 disorder).


Marco Formentin (Padova): Metastates in mean-field models with random external fields generated by Markov chains.

Authors : M. Formentin, C. Kuelske, A. Reichenbachs.

Abstract : We extend the construction of metastates in finite-state mean-field models to situations where the local disorder terms are a sample of an external ergodic Markov chain in equilibrium. We show that for non-degenerate Markov chains, the structure of the theorems is analogous to the case of i.i.d. variables when the limiting weights in the metastate are expressed with the aid of a CLT for the occupation time measure of the chain. As a new phenomenon we also show in a Potts example that for a degenerate non-reversible chain this CLT approximation is not enough, and that the metastate can have less symmetry than the symmetry of the interaction and a Gaussian approximation of disorder fluctuations would suggest.


Patricia Gonçalves (Minho): Additive functionals of exclusion processes.

Authors : P. Gonçalves, M. Jara.

Abstract : In this poster I consider exclusion processes denoted by $(\eta_t)_{t\geq{0}}$, evolving on $\mathbb{Z}$ and starting from the Bernoulli product measure with constant parameter $\rho\in{[0,1]}$. The goal of the work consists in establishing scaling limits of the functional \begin{equation*} \Gamma_t(f):=\int_{0}^t f(\eta_s)ds \end{equation*} for proper local functions $f$. When $f(\eta):=\eta(0)$, the functional $\Gamma_t(f)$ is called the occupation time of the origin. I present a method that was recently introduced in Gonçalves and Jara (10') "Universality of the KPZ equation", from which we derive a local Boltzmann-Gibbs Principle for a general class of exclusion processes. For the occupation time of the origin, this principle says that the functional $\Gamma_t(f)$ is very well approximated to the density of particles. As a consequence from the scaling limits of the density of particles we derive the scaling limits of $\Gamma_t(f)$. As examples I present the symmetric simple exclusion, the mean-zero exclusion and the weakly asymmetric simple exclusion. For the latter, when the asymmetry is strong enough such that the fluctuations of the density of particles are given by the KPZ equation, we establish the limit of $\Gamma_t(f)$ in terms of this solution.

   [Poster]

Kevin Kuoch (Paris): A Contact Process with Competitive Immigration on ${\mathbb Z}^d$.

Abstract : We introduce the "contact process with immigration", which is a basic contact process on ${\mathbb Z}^d$ where we randomly drop competitive individuals on each site in order to curb the growth of the first. Our interests lie in the influence of the immigration parameter: we exhibit a phase transition according to the immigration and we study a corresponding mean-field model to analyze in another way the role of immigration.

   [Poster]

Pierre-Yves Louis (Poitiers): Cluster Variation Method for a PCA related many-body potential.

Authors : P.-Y. Louis, W. Ruszel, C. Spitoni.

Abstract : Our goal is to study the phase diagram of a multi-body potential on the ${\mathbb Z}^2$ lattice. The Gibbs measures with respect to this potential are related to the steady states of a family of reversible Markov processes, which are a class of interacting particle systems with parallel updating depending on some parameters $\beta,h$. To go further the known isotropic attractive and zero magnetic field case, we develop the Cluster Variation Method. According to the choice of a family of basic maximal clusters, we get different approximations of the free energy. One is presented here.
The so-called natural iteration method algorithm is then developed to numerically determine the potential's ground states as minimizers of the approximated free energy.
This is work in progress.


Elena Pulvirenti (Roma): Cluster expansion in the canonical ensemble.

Abstract : We consider a system of particles in the continuum confined in a box in $R^d$ interacting via a tempered and stable pair potential.
We prove the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime.
The convergence is uniform in the volume and in the thermodynamic limit it reproduces Mayer's virial expansion providing an alternative and more direct derivation which avoids the deep combinatorial issues present in the original proof.

   [Poster]

Wioletta Ruszel (Eindhoven and Nijmegen): Sandpile models on random trees .

Abstract : We would like to present some recent results for sandpile models on random trees.

   [Poster]

Michele Salvi (Berlin): A CLT for the Dirichlet Energy of the Random Conductance Model .

Abstract : We consider the Random Conductance Model on ${\mathbb Z}^d$, assigning to each nearest-neighbour edge a non-negative random weight. Given a finite set with prescribed Dirichlet boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linearly growing boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge to a deterministic limit as the box-side tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds.

   [Poster]

Sergio Simonella (Roma): On the validity of the Boltzmann equation for positive short-range potentials.

Abstract : We consider a classical system of point particles interacting by means of a positive and short-range potential. We prove that, in the low-density (Boltzmann-Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [1] (appeared after the well known result of Lanford [2] for hard spheres) and of a recent paper by Gallagher et al [3].
[1] F. King. "BBGKY hierarchy for positive potentials", Ph.D. dissertation, Dept. Mathematics, Univ. California, Berkeley, 1975.
[2] O. Lanford III. "Time evolution of large classical systems", Lecture Notes in Physics, E.J. Moser ed., Springer-Verlag 1975, 1-111.
[3] I. Gallagher, L. Saint Raymond, B. Texier. "From Newton to Boltzmann: the case of short-range potentials", available on hal.archives-ouvertes.fr/docs/00/71/98/92/PDF/BGlimit.pdf, 2012.





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