# Abstracts

#### Claudio Landim (IMPA, Rio de Janeiro)

The evolution among ground states in the ergodic time scale.

We examine the asymptotic evolution of two Markov chains.

We start with the Kawasaki dynamics at inverse temperature $\beta$ for the Ising lattice gas on a two-dimensional square of length $2L+1$ with periodic boundary conditions. We assume that initially the particles form a square of length $n$, which may increase, as well as $L$, with $\beta$. We show that in a proper time scale the particles form almost always a square and that the center of mass of the square evolves as a Brownian motion when the temperature vanishes.

Then, we consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^N_t$, $t \ge 0$, be a continuous time simple random walk on $G_N$ which waits a \emph{mean} $W^N_x$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge 2$, random $d$-regular graphs and the largest component of super-critical Erd\"os-R\'enyi random graphs.

#### Milton Jara (IMPA, Rio de Janeiro)

Abrupt thermalisation and profile cut-off for randomly perturbed dynamical systems.

Consider a continuous-time dynamical system with an hyperbolic fixed point which is a local attractor. Start this dynamical system from a point in the basin of attraction of this local attractor. It is well known that the evolution of this point converges exponentially fast to the fixed point. Now add a small random perturbation to this dynamical system. We show that in the time scale on which the original dynamical system approaches the fixed point, the stochastic dynamics converges in an abrupt way to a distribution concentrated around the fixed point. In the case on which the attractor is global or it is the attractor with minimal energy, we show profile cut-off for the stochastic dynamics as its amplitude vanishes. In the case where there are more than one attractor, we show what is called thermalisation in the language of metastability phenomena.

Joint work with Gerardo Barrera.

#### Robert Jack (Bath)

Large deviations, metastability, and effective interactions

We consider time-averaged measurements (i.e., random variables) in physical systems described by stochastic dynamics. In typical cases of interest, these measurements obey a large-deviation principle with a speed proportional to the averaging time. We show conditioning on trajectories that with non-typical values of these random variables, can lead to the emergence of surprising structure [1,2]. We discuss the characterisation of this structure via the construction of "auxiliary systems" for which it becomes typical [3]. There are also links between this structure and the existence of metastable states in these systems.

[1] R. L. Jack and P. Sollich, J. Phys. A 47, 015003 (2014).

[2] R. L. Jack, I. R. Thompson and P. Sollich, Phys. Rev. Lett. 114, 060601 (2015).

[3] R. L. Jack and P. Sollich, Eur. Phys. J.: Special Topics 224, 2351 (2015). doi:10.1140/epjst/e2015-02416-9

#### Cristina Toninelli (Paris 6-7)

Kinetically constrained spin models: some results and open issues

Kinetically constrained spin models are interacting particle systems with Glauber dynamics. The key feature is that birth and death of particles occur only if the configuration satisfies a local constraint. We will show how the constraints induce the existence of clusters of blocked sites, the occurrence of several invariant measures, non-attractiveness, ergodicity breaking transitions and the failure if classic coercive inequalities to analyze relaxation to equilibrium. We will present some techniques which have been developed to analyze ergodicity, the scaling of spectral gap and mixing times, as well as relaxation to equilibrium. We will conclude by presenting some open problems.

#### Alessandra Faggionato (La Sapienza)

Metastable features in the d-dimensional East model

The East model on Z^d, d>=1, is an example of oriented and cooperative kinetically

constrained spin system. It exhibits several features of glassy dynamics, as slow relaxation,

aging, presence of infinite relevant time scales, heterogeneity,…

In this talk we will concentrate on some metastable aspects of the model. Starting from out-of-equilibrium configurations, before relaxing the system spends much time in regions with high energetic barriers, behaving as traps. Despite classical metastable systems, due to the large number of escape paths, in the East model energetic and entropic contributions are equally relevant in determining mean escape times. Relevant tools to study these phenomena are provided by potential theory.

Based on joint works with P. Chleboun and F. Martinelli.

#### Oriane Blondel (Lyon)

A random walk on the East model.

We consider certain random walks in a dynamic environment given by a Markov process with spectral gap. After deriving general results concerning the asymptotic behaviour of the walker and the invariant measure for the process, we focus on a specific random walk. When the environment is reversible, we find that the speed of the walker displays a non-trivial symmetry. We discuss in particular the case where the environment is given by the East model.

#### Aaron Smith (Ottawa)

Mixing Time of a Kinetically Constrained Process in High Dimensions at Low Density.

The kinetically constrained Ising process (KCIP) was introduced by Fredrickson and Andersen to study the glass transition. I will present recent work on the mixing properties of the KCIP at low density; our main result is a bound on the mixing time of the KCIP on the lattice $\mathbb{Z}_{n}^{d}$ that is tight up to logarithmic factors. I will present two motivations of the model, survey some recent work on related problems, and give a short proof sketch that emphasizes the relationship between the KCIP and other interacting particle systems. Time permitting, I will discuss extensions of these methods to other 'local' interacting particle systems.

#### Shirshendu Ganguly (Washington)

Non-ﬁxation for conservative stochastic dynamics on the line

We consider Activated Random Walk (ARW), a model which generalizes the Stochastic Sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on Z with mass conservation. One starts with a mass density $\mu > 0$ of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate $\lambda > 0$. Sleepy particles become active on coming in contact with other active particles. We investigate the question of ﬁxation/non-ﬁxation of the process and show for small enough $\lambda$, the critical mass density for ﬁxation is strictly less than one, thereby proving the ﬁrst such result in the symmetric setting. We will present several important questions that still remain open.

#### Eial Teomy (Tel Aviv)

Relation between structure and dynamics in kinetically-constrained models.

In kinetically-constrained models there are no interactions between particles, and therefore there are no static correlations and no inherent static structure. However, due to the kinetic constraints, some regions are more mobile than others and thus there are dynamical spatial correlations which are related to the distribution of mobile regions.

We investigate the distribution of the mobile regions by considering the minimum number of particles that need to move before a specific particle can move. This structural property, the culling time, is easy to find by iteratively culling mobile particles from a snapshot of the system. It is the same technique used to find whether the system is ergodic or not.

We compare these structural properties to the dynamics in these models by measuring the persistence time, which is the average time it takes a particle to move for the first time. By mapping the dynamics to diffusion of vacancies we find an algebraic relation between the mean culling time and the persistence time, with a model-dependent exponent.

#### Juan P. Garrahan (Nottingham)

Ideas about quantum metastability.

I will consider the problem of slow relaxation and non-ergodicity in quantum systems. I will address these issues using ideas borrowed from the dynamics of classical glasses, where slow relaxation is often a consequence of dynamical constraints rather than imposed disorder. I will show how such constraints can give rise to slow dynamics and metastability in both closed quantum systems evolving unitarily, and open quantum systems evolving dissipatively.

I will discuss elements of a theory of quantum metastability.

#### Peter Sollich (King's College London)

Driven dynamics of trap models.

Kinetically constrained models have been extensively analysed in terms of biased trajectory ensembles. These probe large deviations of path-dependent observables such as the activity, i.e. the total number of configuration changes over some fixed long time interval, and can be used to understand e.g. dynamical phase transitions. The steady state dynamics in a biased ensemble is still Markovian but with an additional effective energy assigned to each configuration. Even for simple KCMs like the East model this effective energy is generically a complicated many-body interaction.

Here we apply the biased ensemble approach to study trap models from the same point of view as KCMs, driving them to atypically large or small activity. This provides a simple but interesting platform for understanding the interaction of "glassy" aging dynamics - as takes place in unbiased trap models at low temperature - and driving by ensemble bias. In particular, we find that for trap models one has to generalize the effective energy approach, allowing the effective energy to be time-dependent. I will focus on the Bouchaud trap model, which exhibits "energetic" aging, but will comment on commonalities and differences with the entropic aging case (Barrat-Mezard model) if time permits.

#### Takahiro Nemoto (Paris VII)

Universal finite-size scaling for a dynamical phase transition in a kinetically constrained model and a quantum phase transition in a ferromagnet.

Kinetically Constrained Models display a dynamical phase transition in the biased ensemble where trajectories are weighted with respect to their activity. This phase transition is mathematically similar to the one observed in a quantum phase transition of the Ising ferromagnet at zero temperature. In this talk, we show this similarity in a clear manner, and derive a universal scaling function and a scaling gap describing finite-size effects in the mean-field versions of those models.

#### Johel Beltran (PUCP, Lima)

Nucleation phase of condensing zero range process

In this talk we consider a zero range process with N particles on a fixed finite set of sites and whose jump rate is g(n)=1+b/n. As N tends to infinity, we observe the evolution of the fraction of particles at each site in the time scale N^2. Clearly, the state space for such evolution is a simplex E. For b>1, we prove that this sequence of processes converges to a Feller continuous Markov process with continuous paths on the simplex E. Furthermore, this limiting process exhibits an absorbing property: once a coordinate vanishes, it remains zero forever. In particular, each vertex of E turns to be a trapping state for the process. We characterize this process as the unique solution of a martingale problem. In our case, the standard results on uniqueness for diffusion with boundary conditions do not apply directly because the drift diverges as the diffusion approaches the boundary. We shall see that the absorbing property is in fact a consequence of this feature. This is a joint work with Milton Jara and Claudio Landim.

#### Ines Armendariz (Buenos Aires)

Zero Range Process with superlinear rates

We prove the existence of ZRP dynamics with non-decreasing rate $g$ in general dimensions under translation invariant initial conditions, and show invariance of the usual family of product measures parametrized by particle density .

In one dimension and with nearest neighbour dynamics we obtain stronger results: the construction works for initial conditions with finite asymptotic density and the solution satisfies the associated martingale problem. The main tool is coupling, based on the attractiveness of the process.

This is joint work with Enrique Andjel and Milton Jara.

#### Michalis Loulakis (Athens)

Metastability in a condensing zero-range process in the thermodynamic limit

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density by Beltran and Landim, and we follow their martingale approach. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.

This is joint work with Ines Armendariz and Stefan Grosskinsky

#### Sander Dommers (Bochum)

Metastability in the reversible inclusion process.

In the reversible inclusion process with $N$ particles on a finite graph each particle at a site $x$ jumps to site $y$ at rate $(d+\eta_y) r(x,y)$, where $d$ is a diffusion parameter, $\eta_y$ is the number of particles on site $y$ and $r(x,y)$ is the jump rate from $x$ to $y$ of an underlying reversible random walk.

When the diffusion $d \to 0$ as $N \to \infty$, the particles cluster together to form a condensate. It turns out that these condensates only form on the sites where the underlying random walk spends the most time. Once such a condensate is formed the particles stick together and the condensate performs a random walk itself on much longer timescales, which can be seen as metastable behavior.

We study the rates at which the condensate jumps and show that in the reversible case there are several time scales on which these jumps occur depending on how far (in graph distance) the sites are from each other. This generalizes work by Grosskinsky, Redig and Vafayi who study the symmetric case where only one timescale is present. Our analysis is based on the martingale approach by Beltrán and Landim.

This is work in progress jointly with Alessandra Bianchi and Cristian Giardinà.

#### Alessandra Bianchi (Padova)

Metastability in the reversible inclusion process II: Toward the characterization of the relaxation time.

The inclusion process (IP) is a stochastic lattice gas where particles perform random walks subjected to mutual attraction, thus providing the natural bosonic counterpart of the well-studied exclusion process. Due to attractive interaction between the particles, the IP can exhibit a condensation transition, where a positive fraction of all particles concentrates on a single site. In this talk, following the setting and results presented by S. Dommers, we consider the reversible IP on a finite set S, in the limit of total number of particles going to infinity, and address the problem of characterizing the relaxation time of the dynamics. This will connected with the study of the formation of a condensate starting from two half-condensates (nucleation phase) and will lead to the identification of possibly many transition time scales, which are generally longer than that characterizing the dynamics of the condensate on neighboring sites (discussed in Dommers's talk). We approach the problem starting from potential theoretic techniques and following some recent related ideas developed in a few papers we will refer to. Joint work with S. Dommers.

#### Martin Slowik (TU Berlin)

Metastability in stochastic dynamics: Poincar\'e and logarithmic Sobolev inequality via two-scale decomposition.

Metastability is a phenomenon that occurs in the dynamics of a multi-stable non-linear system subject to noise. It is characterized by the existence of multiple, well separated time scales. The talk will be focus on the metastable behavior of reversible Markov chains on countable state spaces. In particular, I will discuss an approach to derive optimal constants in the Poincar\'e and logarithmic Sobolev inequality. The proof is based on a refined two-scale decomposition. A key ingredient in our approach is a discrete analogue of the capacitary inequality of Vladimir Maz'ya leading to estimates that are valid beyond the metastable setting.

#### Elisabetta Scoppola (Roma Tre)

Biking to equilibrium with Probabilistic Cellular Automata

Probabilistic cellular automata (PCA) with self interaction can be used for sampling from Gibbs measure. In the reversible case the construction is quite robust. In the 2d Ising model with periodic boundary condition, a non reversible PCA can be introduced with fast mixing.

#### Emilio Cirillo (La Sapienza)

Exit time in presence of multiple metastable states

In the framework of the Blume Capel model the effect of the presence of multiple metastable states on the way the system exits such states is discussed.

#### Francesca Nardi (Eindhoven)

Metastability for General Dynamics with Rare Transitions: Escape Time and Critical Configurations.

Metastability is a physical phenomenon ubiquitous in first order phase transitions.

A fruitful mathematical way to approach this phenomenon is the study of rare transitions for

Markov chains. For Metropolis chains associated with statistical mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. In the seminar we provide a similar description in the general rare transitions setup. Beside their theoretical content, we believe that our results are a useful tool to approach metastability for non-Metropolis systems such as Probabilistic Cellular Automata.

This is joint work with Emilio N. M. Cirillo, Julien Sohier