MINI COURSES
AND
LONG TALKS

Mini-courses:

Title:  Equilibrium Fluctuations of the SSEP on Random Graphs with Conductances

Abstract: The symmetric simple exclusion process (SSEP) on a generic graph with weights (conductances)  is a fundamental stochastic interacting particle system. It consists of random walkers on the graph subject to the site-exclusion interaction, with jump rates along the edges determined by the conductances. We introduce a unified framework based on group actions to study graphs embedded in R^d that are microscopically disordered but statistically homogeneous under translations. We show how stochastic homogenization and duality play a crucial role in analyzing the large-scale behavior of the SSEP. In particular, we focus  here on the equilibrium density fluctuation field and its convergence to a generalized Ornstein–Uhlenbeck process. While a universal result is obtained for d>=3, the same conclusion holds for d = 2 only under additional regularity assumptions ensuring local Hölder estimates for the associated parabolic equations. In the lectures, we will also review some fundamental results in regularity theory, closely connected with the De Giorgi–Nash–Moser theory for partial differential equations. Finally, we will discuss applications to some classical random graph models.

Title:  Localization transition for directed polymers in a random environment (in dimension larger than 3)

Abstract: The Directed Polymer in a Random Environment (DPRE) is obtained by reweighting the trajectories of finite length simple random walk using an i.i.d. random environment. It is one of the simplest disordered models in statistical mechanics, and one for which the disorder-induced phase transition has been intensively studied. When the intensity of the disorder increases, the systems behavior changes drastically: at high temperature (low disorder intensity) the trajectories of the polymer are diffusive with a behavior which is very similar to that of the simple random walk, at low temperature, the trajectories of the polymer are conjecture to concentrate on a narrow space-time corridor, and its end-point distribution is localized.

In this course, we will present the state of the art for the study of this transition in dimension larger than 3, and expose simple proof of some well established localization and delocalization results.

Long talks:

Title: Limits of massive spanning forests

Abstract: The lambda spanning forest of a graph G is a random subgraph that generalises the notion of the uniform spanning tree. It was recently shown by D'Achille, Enriquez and Melotti that on the complete graph with n vertices, these forests admit a (local) limit as n goes to infinity. This limit can fall into three regimes depending on whether the parameter lambda grows faster than, slower than, or at the same rate as n. We generalise their result to show that the limits hold on any sequence of regular graphs with degrees tending to infinity, provided that lambda is tuned correctly as a function of the degree parameter. 

The proof exploits fundamental links between uniform spanning trees and loop-erased random walks. This joint work with Breki Pálsson (University of Iceland).

Title: Localization induced by Sinai disorder in stochastic exchange models with continuous mass

Abstract: In recent years, much attention has been devoted in both the physics and mathematics literature to studying reversible stochastic exchange models with continuous conserved total mass, generalizing the well-known Kipnis-Marchioro-Presutti (KMP) model. In this talk, we propose a non-reversible variant of such interacting particle systems with bond disorder. On the one hand, this model preserves part of the algebraic and duality structures unveiled for its reversible counterparts; on the other hand, it leads to drastically different equilibrium states and behaviors. We explore the resulting emergent equilibrium mass profile on a 1D lattice with closed boundary conditions, where the underlying disorder induces a mass localization mechanism driven by a Sinai potential. As we will show, the quenched point-mass marginal at distance k from the boundary is governed by a polymer measure, characterized by the exponential of the position of a standard symmetric random walk on the integers after k steps. At infinite temperature, this polymer measure degenerates into the uniform steady-state masses of the standard KMP model, whereas at low temperatures, it gives rise to strong localization phenomena related to the extremes of the symmetric random walk path and its associated functionals.  We will in particular discuss the scaling limits of the masses at a fixed distance from the boundaries, the appearance of certain associated arcsine laws, and the limiting laws of the extremal masses. 

Ongoing joint work with Michele Giusfredi e Paolo Politi. 

Title: Some properties of 2D directed polymers and Stochastic Heat Flow

Abstract: I will review some recent results on the directed polymer model in dimension d=2, which can be seen as a discretised version of the stochastic heat equation. Recent results have shown that, in a suitable (and subtle) critical window of parameters, the model converges to a disordered limit, namely the Critical 2D Stochastic Heat Flow (SHF) introduced by Caravenna, Sun and Zygouras. This SHF is a non-Gaussian stochastic process of measures on R^2, and a lot of its properties have been studied over the last years. In this talk, I will first take some time to explain the (idea of the) construction of the SHF and then review some of the recent advances. In particular, I will comment on recent/ongoing works in collaboration with Caravenna, Turchi and Zygouras, which study how the mass assigned by the SHF to a ball behaves, either as time goes to infinity or as the radius of the ball goes to 0.

Title: A stochastic homogeneization approach to supercritical SPDEs

Abstract: In the present talk, we will present a novel approach to study the large-scale fluctuations of a class of stationary quasi-linear SPDEs at super-critical dimensions. In particular, we show that this class of SPDEs has Gaussian fluctuations and that the limit is given by a linear additive Stochastic Heat equation with renormalised coefficients. Our approach is based on a suitable two-scale expansion for the generator of our dynamics and extends classical stochastic homogeneization to the infinite-dimensional setting. Byproducts of our methods are quantitative rates for the convergence of the microscopic generator to the limiting one and a more transparent characterisation of the macroscopic diffusivity in terms of microscopic observables.
Based on joint ongoing work with H. Giles (TU Vienna) and L. Gräfner (Warwick). 

Title:  Non-Equilibrium Steady State, Large Deviations and Shannon Entropy of Harmonic Models

Abstract: In this talk, we consider the class of Harmonic models on a 1D lattice with open boundaries, for which large amounts of mass can be transferred in a single jump. We show that the non-equilibrium steady state can be explicitly characterised via a mixture of inhomogeneous product measures, with mixing parameters given by the order statistics of uniform random variables. This structure allows to compute the macroscopic large deviation function and to establish an additivity principle. Furthermore, we analyse the steady-state Shannon entropy. We show that the leading O(N) term coincides with the one of local equilibrium, while the first O(1) correction depends only on the rescaled two-point truncated correlation and matches the one of Brownian bridge as N→∞.

This is from joint works with: Chiara Franceschini, Rouven Frassek, Davide Gabrielli, Cristian Giardinà, Frank Redig, Dimitrios Tsagkarogiannis.

Title: Towards a phase transition for the integer-valued membrane model

Abstract: In this talk we study the integer-valued membrane model on a finite box of the integer lattice at inverse temperature \beta. This is a random interface with bilaplacian energy constrained to the integers. For any temperature and any test function, the model has sub-Gaussian upper bounds on the tails. In dimensions 2\le d\le 4 we prove matching lower bounds for a broad class of localized test functions in a high-temperature regime in which the inverse temperature is allowed to depend on the box size and as such we identify a smallness condition on the inverse temperature under which the lower bound holds. Joint work with B. Dan (IISER Kolkata) and R. S. Hazra (Leiden University).

Title: Uniqueness of the invariant measure for the phi42 dynamics 

Abstract: I will discuss the phi42 dynamics in infinite volume, a singular SPDE on the full plane, and study its invariant measures. In finite volume those are known to be unique and given by the phi42 field theory. In infinite volume uniqueness may fail, for instance due to phase transitions. We prove uniqueness whenever the susceptibility of the finite volume phi42 measure remains bounded, a presumably optimal criterion. This is done by adapting to the field-theory setting the Holley-Stroock-Zegarlinski approach to uniqueness for statistical mechanics models. This approach is based on a volume-independent bound on the log-Sobolev constant of the associated dynamics, together with crude, model-independent bounds. In the phi42 case the log-Sobolev has been established, but substantial work is required to adapt the other parts of the argument. I will spend some time describing the Holley-Stroock-Zegarlinski argument in detail and try to give some intuitions of the difficulties when trying to adapt it to the field-theory setting.
The talk is based on joint work with R. Bauerschmidt and H. Weber.

Title: Two phase transitions for catalytic branching Markov chains

Abstract: Consider a continuous-time branching Markov chain (Z_t, t ≥ 0) on a locally finite graph G rooted at o. Each particle moves according to an irreducible Markov process ξ and branches at a rate that depends on their location: the branching rate is λ_o ≥ 0 at the root and λ ≥ 0 elsewhere. The offspring distribution is concentrated on {1, 2, . . .}, with mean m > 1, and finite second moment. We characterize the recurrence/transience phase transition for this catalytic branching Markov chain. Furthermore, under suitable assumptions we prove a second phase transition concerning the "local-to-global" ratio in the population. When the graph is the integer lattice G = Z^d and ξ is the simple random walk, our results confirm several conjectures of Mailler and Schapira [Ann. Appl. Probab. 2026], which studied this model via a different approach.
Based on joint work work with Xinxin Chen, Haojie Hou and Quan Shi

Title: Extremes of the zero-average Gaussian Free Field on random regular graphs  

Abstract: We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random r-regular graphs and the Gaussian free field on r-regular trees. For random r-regular graphs of diverging size, for every fixed r≥3, we show that, with high probability, the rescaled extremal point process of the field is asymptotically distributed as a Poisson point process on the line with intensity e^{-x}dx. The same limit behaviour is obeyed by the restriction of the GFF on r-regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.
This talk is based on joint work with Andreas Klippel and Christian Mönch.  

Title: From 1d random Schrödinger operators to random Dirac operators

Abstract: We will consider random Schrödinger operators in 1d, both in the discrete and the continuum setting. It is known that for a large class of potentials, Anderson localisation holds: the spectrum is pure point and the eigenfunctions are exponentially localized. In the weak disorder limit, we will show that a random Dirac operator arises « generically ». This operator displays very nice properties.
Joint work with Laure Dumaz (ENS). 

Title: Nonequilibrium fluctuations of interacting particle systems.

 
Abstract: We review some recent results on nonequilibrium fluctuations of gradient and reaction-diffusion models. 

Title: Topology, Noise, and Trapping in Circular Opinion Dynamics

Abstract: In this talk we will explain stochastic opinion dynamics with opinions taking values on the circle. We consider a local midpoint averaging dynamics on path and ring graphs. Although the update rule is locally contractive and consensus is the unique absorbing state for finite systems, periodic boundary conditions introduce an integer-valued winding number that partitions the configuration space into topological sectors and profoundly modifies the transient dynamics. We will explain how branch-crossing events provide the unique mechanism for changing the winding number and how a lifted representation together with an adaptive co-moving frame leads to a rigorous contraction result within a fixed winding sector.We will then discuss recent extensions of the model incorporating bi-modal noise and parallel updates. Numerical experiments show how these mechanisms affect the stability of winding configurations, generate collective spin-flop behavior, and motivate new macroscopic observables that distinguish synchronization, symmetry breaking, and topological organization.

This is joint work with Cristian Spitoni (UU) and Annika Brockhaus (U Leiden).

Title:  The periodic directed landscape  

Abstract: The Kardar--Parisi--Zhang universality class is a central topic in mathematical physics and probability theory. In this talk, we discuss the conjectural scaling limit of periodic models in the Kardar--Parisi--Zhang universality class: the periodic directed landscape. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape and give a variational characterization of the periodic KPZ fixed point, recently constructed by Baik, Liao, and Liu. We also establish the convergence of periodic asymmetric simple exclusion processes to periodic KPZ fixed points, coupled through the same periodic directed landscape. A key ingredient is a gluing technique for combining overlapping directed landscapes, which may be of independent interest.
This is based on joint work with Amol Aggarwal and Ivan Corwin. 

Title: Uniqueness of the infinite cluster for monotone percolation models without insertion tolerance

Abstract: We will study the infinite cluster of toppled sites during the stabilization of the Abelian sandpile model. We will show that in the supercritical regime the infinite cluster is unique almost surely. The main challenge in this model is that it does not have the insertion tolerance property.

Based on a joint work with Christoforos Panagiotis (Univ. of Bath)

Title: Fluctuation and big-jump phenomena in the localized phase of the pinning model 

Abstract: The pinning model is an elementary statistical mechanics model with disorder that exhibits a localization transition. This model consists of a Gibbs change of measure of a renewal process that naturally arises in the modeling of a variety of phenomena in physics and biophysics. In this talk, I review recent progress in the study of the localized phase of the model, notably the regularity class of the free energy, the scaling of the largest excursion between pinned sites, and the central limit theorems for both the contact number and its mean. I also discuss the occurrence of a big-jump phenomenon as the mechanism behind the large deviations of the contact number in the homogeneous model. The presence of disorder has a drastic effect on this phenomenon, which in fact disappears as soon as even weak disorder is introduced.