We consider a simple multi layer ferromagnetic Ising model consisting of two mean-field layers, subject to external magnetic fields h_1 and h_2, and coupled through a local ferromagnetic inter-layer interaction J. 

We first discuss the zero-temperature regime, where the phase diagram can be characterized explicitly in terms of the competition between the external fields and the inter-layer coupling. We then investigate the low-temperature regime and show that, for sufficiently large $\beta$, the phase boundaries of the zero-temperature diagram persist and can be continued to positive temperatures.

Although the model is minimal, it already captures several non-trivial features that arise in multi-layer systems. In particular, it exhibits a rich interplay between the tendency of the two layers to synchronize and the asymmetry induced by the external fields. One of the main outcomes of the analysis is the thermal stability of the zero-temperature phase boundaries, which persist for sufficiently large values of $\beta$. 

These results may serve as a first step toward the analysis of more realistic multi-layer systems with finite-range and intra-layer interactions.

In this talk, we introduce the spherical ensemble and more generally, the ratio of two independent random matrices with i.i.d. entries. We begin by reviewing some previous results for a single random matrix with i.i.d. entries. We then show that, for any test function with finitely many logarithmic singularities, the linear statistics of the spherical ensemble converge to a Gaussian distribution, after a suitable normalization by 1/\sqrt{log n}. The limiting distribution depends only on the weights of singularities. As an application, we obtain the finite-dimensional Gaussian convergence of the logarithm of the characteristic polynomial, normalization by 1/\sqrt{log n}. Moreover, we explicitly compute the variance and covariance without the 1/\sqrt{log n} normalization, showing that the field is log-correlated. All these results extend beyond spherical ensemble under a four moment matching condition. 

This is based on joint work with Djalil Chafaï and David García-Zelada.

In the context of dependent random fields, we study the relationship between whether a field satisfies Gaussian concentration bounds (classically known as McDiarmid inequalities for independent processes) and whether it can be obtained as a finitary coding of an i.i.d. random field. In particular, we show that Gaussian concentration holds whenever the coding volume has a finite second moment. For classical examples like the Ising or Potts models, this approach yields sharp necessary and sufficient conditions, proving that Gaussian concentration holds if and only if the model lies in the full uniqueness regime. We further apply these results to a broad class of processes, ranging from Gibbs measures and Markov random fields on Zd to various one-dimensional processes. 

We derive a scaling limit in law as N tends to infinity for the cover time by a simple random walk of the subgraph of the square lattice obtained by discretizing an N-scale blow up of a planar domain and adding a wired boundary to it. The limiting distribution is that of a Gumbel random variable shifted by an independent (random) quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the rescaled location of the last visited vertex by the walk. Here the limit turns out to be precisely the (expected) critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well-known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field and, in particular, with that of the discrete Gaussian Free Field conditioned to have zero average. 

Joint work with Oren Louidor (Technion).

We introduce a model of epidemics among moving particles on any locally finite graph. At any time, each vertex is empty, occupied by a healthy particle, or occupied by an infected particle. Infected particles recover at rate $1$ and transmit the infection to healthy particles at neighboring vertices at rate $\lambda$. In addition, particles perform an interchange process with rate $\mathsf{v}$, that is, the states of adjacent vertices are swapped independently at rate $\mathsf{v}$, allowing the infection to spread also through the movement of infected particles. On $\mathbb{Z}^d$, we start with a single infected particle at the origin and with all the other vertices independently occupied by a healthy particle with probability $p$ or empty with probability $1-p$. We define $\lambda_c(\mathsf{v}, p)$ as the threshold value for $\lambda$ above which the infection persists with positive probability and analyze its asymptotic behavior as $\mathsf{v} \to \infty$ or $\mathsf{v} \to 0$, for fixed $p$.  

Joint work with Marcelo Hilário (UFMG), Maria Eulália Vares (UFRJ) and Daniel Valesin (University of Warwick).

Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. We study the relaxation time of the associated Glauber dynamics, proving its conjectured super-Arrhenius scaling at low temperature. We also reveal highly unusual behaviour in finite volume around the critical length scale.

The talk is based on joint work with Laurent Bartholdi and Ivan Mitrofanov.

The study of the cut-off phenomenon—also known as abrupt convergence (to equilibrium) or abrupt thermalization—has a history spanning several decades, from the seminal work of Aldous and Diaconis (1986) [1] on random card shuffling to its appearance in many modern contexts. In very simplified terms, the cut-off phenomenon refers to the following behavior: starting from a fixed initial distribution (typically a Dirac delta), one evolves under a given dynamics.

When measuring the distance between the distribution at time t and the stationary distribution (in some preferred classical metric), one observes that this distance remains essentially maximal for a long time and then drops abruptly to zero on a specific time scale, staying close to zero thereafter.

In our work, we introduce and study a phenomenon that, to the best of our knowledge, has not been previously analyzed on its own within the rigorous mathematical literature, and which we call abrupt decorrelation. We start from a random initial distribution m_0. As the dynamics evolves, we observe that the correlation between the distribution at time t, denoted by m_t and the initial distribution m0, undergoes an abrupt decay. More explicitly, we consider the distance (in some classical metric) between the joint law of (m_0,m_t) and the product measure m_0 × m_t in order to detect this phenomenon. In an analogous way as the theoretical framework for the cut-off phenomenon developed by Barrera and Ycart (2014) [2], we propose three levels of abrupt decorrelation (at a sequence of times, with a window, and with a profile). In this talk, we focus on a particularly tractable case where many explicit computations are possible: Ornstein–Uhlenbeck processes.

Joint work with Leandro Pimentel and Juan Carlos Pardo.

References:

[1] Aldous, D., Diaconis, P. Shuffling cards and stopping times. Amer. Math. Monthly 93 no. 5, 333–348 (1986).

[2] Barrera, J., Ycart, B. Bounds for left and right window cutoffs. ALEA Lat. Am. J. Probab. Math. Stat. 11, no. 2, 445–458, (2014).

We investigate rumor spreading in a generalized Maki-Thompson model with spontaneous stifling, evolving on quasi-transitive networks. Individuals are either ignorants, spreaders, or stiflers; spreaders stop by contact with other spreaders or stiflers or after an independent random waiting time sampled from a given distribution, modeling a spontaneous loss of interest. The topology of the underlying population network is incorporated by modeling it as a broad class of symmetric networks, whose vertices are partitioned into finitely many orbit types. This yields a unified framework for homogeneous and heterogeneous networks. For sequences of finite quasi-transitive graphs, and for infinite quasi-transitive graphs with subexponential growth, we establish a Functional Law of Large Numbers and a Functional Central Limit Theorem for the densities of each vertex type for the three states. The mean-field limit is described by a system of nonlinear integral equations, while fluctuations are asymptotically Gaussian and governed by a system of stochastic integral equations with explicit covariance. Our results show how the topology and the law of spontaneous stifling jointly shape the speed and variability of rumor outbreaks. As a special case, our model reduces to the classical Maki-Thompson model when spontaneous stifling is absent.

We explore near-critical Bernoulli percolation on the first quadrant of the square lattice. Columns and rows are selected based on the arrivals of a process given by $i.i.d.$ copies of some random variable $\xi$. Vertices lying at the intersection of a selected row and a selected column are open with probability $p$, whereas the remaining vertices are open with probability $q$. We investigate the near-critical regime $p=p_c+\varepsilon$ and $q=p_c-\delta$, where $p_c$ denotes the critical threshold for homogeneous site percolation. We prove that the effect of the reinforcement depends on the tail behavior of the spacing variables. For geometric spacings, the critical threshold is strictly shifted, and this shift persists under additional independent dilution of the environment. For polynomially decaying tails, the critical threshold remains $p_c$, although a first-order phase transition occurs. We also show that, for any $\eps>0$, percolation does not occur at $p=p_c+\eps$ and $q=p_c$ when the spacings decay slower than exponentially in one direction.

The talk is based on joint work with Estevão Borel, Marcos Sá, and Rémy Sanchis. 

In this talk, we consider a particle that moves linearly inside generalised parabolic domains and undergoes random reflection upon hitting the boundary. The aim is to determine, as a function of the reflection law, in which geometric regimes the process is recurrent or transient.

This ongoing work addresses the non-elliptic case, in which the reflection law may lose uniformity and exhibit polynomial tails. The central idea is to relate the problem to a class of non-homogeneous random walks and to study a suitably rescaled version of the process through the construction of a Lyapunov function. From this point on, we use supermartingale methods to identify, in terms of the reflection kernel, a phase transition between recurrence and transience as a single geometric parameter of the domain is varied.

This ongoing joint work with Niels Kohlenbrander (Leiden University) continues a line of research developed in collaboration with M. V. Menshikov and A. R. Wade (Durham University) on stochastic billiards with Markovian reflections in generalised parabolic domains. Here, we address the non-elliptic case, in which the reflection law may lose uniformity and exhibit polynomial tails.

In collaboration with Dirk Erhard, we are interested on quantitative theorems for the fraction of opinions one of the noisy voter model. Aljovin, Jara, Xiang (2024) proved a quantitative law of large numbers when the underlying graph is the (deterministic) complete graph. In this talk we discuss the effects of the randomness coming from the graph.

In this talk we introduce a class of cellular automata growth models on the two-dimensional integer lattice with finite cross neighborhoods. These dynamics are determined by a Young diagram $Z$ and the radius ρ of the neighborhood, which we assume to be sufficiently large, so that the resulting dynamics are supercritical. A point becomes occupied if the pair of counts of currently occupied points in the horizontal and vertical parts of the cross neighborhood lies outside $Z$. Starting with a small density p of occupied points, we focus on the first time T at which the origin is occupied. We consider the cases when $Z$ is a triangle, a rectangle, and the union of a finite rectangle with an infinite strip. The distinguishing feature of these dynamics is nucleation of lines that grow to significant length before most of the space is covered.

This is joint work with J. Gravner, D. Sivakoff, and L. Wilson.

Interacting particle systems are a central topic in probability theory, often studied in discrete state spaces such as lattices or graphs. In this talk, I will discuss a continuous counterpart, where particles move in Euclidean space according to stochastic differential equations and interact through possibly singular forces acting in the drift. As the number of particles tends to infinity, the empirical measure converges as a process to the solution of a stochastic Fokker-Planck equation, describing the collective evolution of the particle cloud. I will present quantitative estimates for this convergence, providing explicit moment bounds for the approximation error between the particle system and the limiting stochastic evolution. In particular, the results yield rates of convergence at the process level, rather than merely convergence in distribution.

The zero range process is a classical interacting particle system whose macroscopic behaviour is governed by nonlinear diffusion equations. In a seminal work, Ferrari, Presutti and Vares proved that non-equilibrium density fluctuations converge to a generalised Ornstein-Uhlenbeck process, i.e. a linear stochastic PDE with coefficients determined by the hydrodynamic limit.

In this talk, I revisit these results from the perspective of fluctuating hydrodynamics. I explain how the limiting fluctuation equation can be interpreted as the linearisation of the Dean-Kawasaki equation around the macroscopic density profile. I then discuss recent progress on regularised Dean-Kawasaki equations, including well-posedness results and small noise fluctuation theory, and how these provide a nonlinear SPDE framework extending the classical fluctuation results.

The results are based on the paper https://doi.org/10.1016/j.spa.2024.104503
and preprint https://doi.org/10.48550/arXiv.2504.17094.

In this talk we consider a specific variant of the classical random recursive tree dynamics, where a vertex at time n+1 has information only on those vertices that have arrived in the interval [j(n),n] for a given sequence j(n)↑∞, and connects to vertices uniformly at random amongst this set. We consider two different regimes on the density information, termed macroscopic and mesoscopic regimes, which respectively correspond to j(n)=θn for some θ∈(0,1), and j(n)=n−n^{β} for some β∈(0,1). Our main interest is in studying asymptotics of various local and global functionals of the network. Our results include a change in behaviour from logarithmic to polynomial for the height of the tree from the macro to the mesoscopic regime. Additionally, we discuss the scaling limit of the tree in the mesoscopic regime, showing that this limiting object experiences a phase transition at β=1/2.

Based on joint work with: Omer Angel, Shankar Bhamidi, Serte Donderwinkel and Akshay Sakanaveeti.

We study a critical multitype Bellman-Harris branching particle system in \(\mathbb R^N\) with a finite type space \(\mathbf K=\{1,\dots,K\}\). Particles of type \(i\) move according to a symmetric \(\alpha_i\)-stable process, have non-arithmetic lifetimes with finite mean, and reproduce according to a critical offspring law whose mean matrix is irreducible and stochastic. The branching mechanism is assumed to be in the domain of attraction of a \((1+\beta)\)-stable law, with \(\beta\in(0,1]\). We prove that the particle system converges, as \(t\to\infty\), to a limiting random measure which is nonzero. We also show that the limiting population preserves the initial intensity measure. Thus, the system persists with full intensity. These results complement the local extinction results of Kevei and Lopez-Mimbela (2011) for critical multitype Bellman--Harris systems.