17 et 18 septembre 2015 - Université de Rouen

## Jeudi 17 septembre

10h45-11h00 : Ouverture officielle
11h00-12h30 : Frank Redig - Duality and exactly solvable models in non-equilibrium (Part I).
12h30-14h00 : Déjeuner
14h00-14h35 : Irène Marcovici - Percolation games, probabilistic cellular automata, and the hard-core model.
14h35-15h10 : Nina Gantert - Biased random walk among random conductances.
15h10-15h25 : Pause café
15h25-16h55 : Frank Redig - Duality and exactly solvable models in non-equilibrium (Part II).
16h55-17h10 : Pause café
17h10-17h45 : Arnaud Rousselle - Quenched invariance principle for random walks on Poisson-Delaunay triangulations.
17h45-18h20$\$:$\$Oriane Blondel - Random walk on environments with spectral gap. Perturbations of Markov processes and applications.

## Vendredi 18 septembre

09h00-10h30 : Frank Redig - Duality and exactly solvable models in non-equilibrium (Part III).
10h30-10h45 : Pause café
10h45-11h20 : Gioia Carinci - From the quantum Lie algebra $U_q(Sl_2)$ to the ASEP $(q,j)$.
11h20-11h55 : Kevin Kuoch - Ergodic theory of the simple inclusion process.
12h00-13h45 : Déjeuner
13h45-14h20$\$:$\$Eric Luçon - Disorder-induced traveling waves in the quenched Kuramoto model.
14h20-14h55 : Vu-Lan Nguyen - End point localization in log gamma polymer model.
14h55-15h30$\$:$\$Clément Erignoux - Modelling of collective dynamics : Hydrodynamics of a non-gradient spin process.
15h30-15h45 : Pause café
15h45-16h20 : Alexandre Lazarescu - Hydrodynamic spectrum of one-dimensional bulk-driven particle gases.
16h20-16h55 : Christophe Bahadoran - Hydrodynamics and quenched local equilibrium for disordered asymmetric zero-range process. (Joint with T. Mountford, K. Ravishankar end Ellen Saada)

## Mini-cours

Frank Redig (Delft University of Technology) Duality and exactly solvable models in non-equilibrium. Duality is one the powerful methods in interacting particle systems. Recently we introduced a Lie algebraic approach that on one hand explains existing dualities such as for the symmetric exclusion process (SEP), and for stochastic models of heat conduction (KMP), and on the other hand gives a constructive machinery to build new models. For classical Lie agebras this method gives symmetric processes, and in order to build the corresponding asymmetric processes, one has to pass to the corresponding deformed algebras. We will explain this approach from a variety of concrete model examples such as SEP, ASEP, SIP, ASIP, KMP etc.
There will be three parts:
1. Introduction to stochastic models of heat conduction: Brownian momentum process, Brownian energy process, KMP process, and their duals: symmetric inclusion process (SIP).
2. Dualities and symmetries; we first show the connection between self duality functions and operators that commute with the generator (symmetries). We then show how the generator L of SIP can be constructed from Lie algebraic ingredients, i.e., from a coproduct applied to the Casimir operator of SU(1,1). This by construction implies that L has many commuting operators. This illustrates a general constructive procedure for processes with many symmetries, and hence self-dualities. We illustrate how this applies also for exclusion process and related models by going from SU(1,1) to SU(2).
3. Construction of asymmetric processes with symmetries and self-dualities. We then show how the formalism of the second item can be implemented for quantum Lie algebras, such as q-deformation of SU(2) (gives asymmetric exlcusion and generalizations) and SU(1,1) which gives ASIP and an asymmetric KMP process.

## Conférences

Christophe Bahadoran (Université Blaise Pascal, Clermont Ferrand) Hydrodynamics and quenched local equilibrium for disordered asymmetric zero-range process. (Joint with T. Mountford, K. Ravishankar end Ellen Saada) We consider the asymmetric zero-range process on the line with site-disordered jump rates This model is known to exhibit a phase transition (Ferrari-Krug 1996) if the disorder distribution has a smooth enough tail around its minimum (which corresponds to slowest sites), in which case there is a critical density above which no invariant measure exists. The hydrodynamic limit for this process was studied by Benjamini and al. (1996) in the subcritical regime. Krug and Sepp\"alainen (2000) extended the hydrodynamic limit to the supercritical regime only in the case of total asymmetry and constant jump rates. On the other hand, local equilibrium results are well-known (Landim (1993)) for the tranlsation invariant process, but are missing in the disordered case.
In this work, for general zero-range processes, we establish the hydrodynamic limit including the supercritical regime, the quenched local equilibrium in the subcritical regime, and show that the local equilibrium in the supercritical regime is modified and sees only the critical density. The latter phenomenon is a space-time version of the loss of mass due to Bose-Einstein condensation (Ferrai-Sisko 2007, Andjel et al. 2000, Bahadoran et al. 2015).

Oriane Blondel (Université Claude Bernard Lyon 1) Random walk on environments with spectral gap.
Perturbations of Markov processes and applications.
We consider random walks in random environments with positive spectral gap. Under suitable assumptions on the "asymmetry" of the jump rates w.r.t. the reference measure, we identify the invariant measure for the process seen from the walker and establish a law of large numbers and invariance principle for the walker. These findings rely on a general result concerning L^2 perturbations of Markov generators. We have in mind applications to random walks on a toy model for glassy systems.
Joint work with Luca Avena and Alessandra Faggionato.

Gioia Carinci (Università di Modena e Reggio Emilia) From the quantum Lie algebra $U_q(Sl_2)$ to the ASEP $(q,j)$. Stochastic duality is a key tool that allows to study an interacting particle system in terms of a finite number of dual particles. In many cases duality can be traced back to the underlying algebraic structure of the system. In this talk I will present a new constructive algebraic approach to self-duality based on the link with representation theory of Lie algebras. In particular I will show how the general scheme has been implemented in the case of the $U_q(Sl_2)$ Lie algebra, to construct a new self-dual process, the ASEP $(q,j)$, that is an extension of the standard Asymmetric Exclusion Process to a situation where sites can accommodate more than one (namely $2j$) particles per site. The process is constructed from a ($2j+1$)-dimensional representation of a quantum Hamiltonian with $U_q(Sl_2)$ invariance by applying a suitable ground-state transformation.

Clément Erignoux (École Polytechnique) Modelling of collective dynamics : Hydrodynamics of a non-gradient spin process. Extensive work has been put in the modelling of collective dynamics in the last decades, building on the work of Viscek&Al (1995). However, most of the theoretical background in collective dynamics modelling relies on mean-field approximations. I will present a lattice model where interactions between partices happen at a purely microscopic level, and describe some of the challenges in the proof of its hydrodynamic limit.

Nina Gantert (Technische Universität München, Germany) Biased random walk among random conductances. For a biased random walk among random conductances, we discuss the Einstein relation as well as the monotonicity of the speed as a function of the bias.
We consider the effective diffusivity (i.e. the covariance matrix in the central limit theorem) of a random walk among random conductances. It is interesting and non-trivial to describe this diffusivity in terms of the law of the conductances. The Einstein relation gives a different interpretation of the effective diffusivity as mobility. The mobility measures the response of the diffusing particle to a constant exterior force: Consider the perturbed process obtained by imposing a constant drift of strength $\lambda$ in some fixed direction. The perturbed motion satisfies (as one can show in many examples) a law of large numbers with effective velocity $v(\lambda)$. The mobility is the derivative of $v(\lambda)$ as $\lambda$ goes to 0. The Einstein relation says that the mobility and the diffusivity of a particle coincide. The Einstein relation is conjectured to hold for a variety of models, but it is proved insofar only for particular cases. We explain how it follows from an expansion of the invariant measures for the environment, seen from the particle.
The talk is based on joint work with Jan Nagel and Xiaoqin Guo.

Kevin Kuoch (University of Groningen, Netherlands) Ergodic theory of the simple inclusion process. The simple inclusion process (SIP) is an interacting particle system where particles perform nearest-neighbour jumps according to a simple symmetric random walk and interact by an "inclusion rule", that is, any particle invites a neighbouring particle to come over its site at rate 1. In the purpose the ergodic theory of the SIP, we show the existence of a successful coupling for a finite number of SIP-particles. As a consequence, we characterise the ergodic measures and their attractors. This is a joint work with Prof F.H.J. Redig (TU Delft).

Alexandre Lazarescu (KU Leuven) Hydrodynamic spectrum of one-dimensional bulk-driven particle gases. Interacting particle gases are often used as toy models to gain insight on more complex systems, especially when out of equilibrium, where a general theoretical framework (such as an equivalent of the free energy) is not available. There are two main ways to approach them analytically: either one can solve them exactly, or one may propose an effective coarse-grained description in the hope that it captures the essential features of the model. The first approach is in principle powerful but very restrictive, and understanding how the second (more versatile) approach can be attempted systematically is a major problem in modern statistical physics. An important step in that direction was achieved through the so-called Macroscopic Fluctuation Theory (MFT), which gives a systematic expression of the large deviation function of hydrodynamic currents and densities for diffusive (boundary-driven or weakly bulk-driven) models. In this talk, we will conjecture that this same formalism can be applied to bulk-driven models through a non-rigorous trick, yielding, for instance, the large deviation function of the current flowing through the system, with a few important differences, which make those models arguably richer than diffusive ones: first, information can be easily obtained on a large family of excited states as well as on the steady state, which is not the case, or at least more difficult, for diffusive systems ; and secondly, one can find regimes where this approach breaks down, leading to dynamical phase transitions which are not present in diffusive systems. All of these conjectures will be checked against the asymmetric simple exclusion process (ASEP), which is solvable exactly.

Eric Luçon (Université Paris Descartes) Disorder-induced traveling waves in the quenched Kuramoto model. This is joint work with Christophe Poquet (Roma).
We will discuss the long time behavior of the stochastic Kuramoto synchronization model with disorder. The Kuramoto model consists in a system of N mean-field interacting diffusions on the circle (rotators), with quenched inhomogeneous frequencies. On a finite time scale [0, T], the Kuramoto model is known to be self-averaging: the empirical measure of the system converges as $N\to\infty$ to the deterministic solution of a nonlinear Fokker-Planck equation which exhibits a stable manifold of stationary solutions (at least when the interaction between rotators is strong enough). The purpose of the talk is to show that, on longer times scales (of order $\sqrt N$), the finite-size fluctuations of the quenched disorder compete with the noise and makes the system deviate from this mean-field behavior: we prove the existence of traveling-waves along the stationary manifold, generated by the asymmetry of the quenched disorder.

Irène Marcovici (Institut Elie Cartan, Nancy) Percolation games, probabilistic cellular automata, and the hard-core model. Let each site of the grid ${\mathbb Z}^2$ be closed with probability $p$ and open with probability $1-p$, independently for different sites. Consider the following two-player game: a token starts at the origin, and a move consists of moving the token from its current site $x$ to an open site in $\{x+(0,1),x+(1,0)\}$; if both these sites are closed, then the player to move loses the game. Is there positive probability that the game is drawn with best play -- i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain one-dimensional probabilistic cellular automaton (PCA), which has already been studied from several perspectives, for example in the enumeration of directed animals in combinatorics, in relation to the golden-mean subshift in symbolic dynamics, and in the context of the hard-core model in statistical physics. In a joint work with J. B. Martin and A. E. Holroyd, we prove that the PCA is ergodic for all p, and that no draws occur for the game on $\mathbb Z^2$. I will present this result, together with some extensions to higher dimensional games.

Vu-Lan Nguyen (Université Paris Diderot) End point localization in log gamma polymer model. As a general fact, directed polymers in random environment are localized in the so-called strong disorder phase. In this talk, based on a joint work with Francis Comets, we will consider the exctly solvable model with log-gamma environment, introduced recently by Seppalainen. For the stationary model and the point-to-line version, the localization can be expressed as the trapping of the end-point in a potential given by an independent random walk.

Arnaud Rousselle (Institut de Mathématiques de Bourgogne, Dijon) Quenched invariance principle for random walks on Poisson-Delaunay triangulations. The Voronoi tiling of an infinite locally finite subset $\xi$ of $\mathbf{R}^d$ is the collection of the Voronoi cells: $$\operatorname{Vor}_\xi(x)=\left\{y\in\mathbf{R}^d:\Vert y-x\Vert\leq\Vert y-x'\Vert,\forall x'\in\xi\right\},\qquad x\in\xi.$$ The associated Delaunay triangulation is its dual graph in which there is an edge between vertices $x$ et $x'$ if $\operatorname{Vor}_\xi(x)$ and $\operatorname{Vor}_\xi(x)$ share a $(d-1)$-dimensional face. When $\xi$ is distributed according to a Poisson point process, this graph is called Poisson-Delaunay triangulation. These realistic models found applications in various fields such that geography, biology or telecommunications. In this talk, we present a quenched invariance principle for the variable speed nearest-neighbor random walk $(X_t)_{t\geq 0}$ on the Delaunay triangulation of a realization $\xi$ of a Poisson point process, that is for the Markov process with generator: $$\label{EqDefGene} \mathcal{L}^\xi f(x):=\sum_{y\in\xi}\mathbf{1}_{y\sim x}\left(f(y)-f(x)\right),\qquad x\in\xi.$$ In other words, we show that, for a.e. realization $\xi$ of the point process and all starting point $x\in\xi$, the rescaled process $(X^\varepsilon_t)_{t\geq 0}=(\varepsilon X_{\varepsilon^{-2}t})_{t\geq 0}$ converges in law as $\varepsilon$ tends to $0$ to a nondegenerate Brownian motion under quenched law.