Abstracts

We introduce and discuss the "harmonic model", an integrable Markov process (arising from the integrable XXX chain with non-compact spins) which describes heat conduction in interacting particle systems [1]. The model is closely related to the celebrated Kipnis-Marchioro-Presutti (KMP) process, which instead arises from the quadratic Casimir. In particular, the two models share an $\mathfrak{sl}_2$ symmetry, which underlies their dualities, although they differ in their energy redistribution rule. Furthermore, the harmonic model is integrable in the sense of Yang-Baxter. Exploiting this structure, we compute the non-equilibrium steady state of the boundary-driven harmonic model in closed form [2]. We show that it can be expressed as a mixture of product Gibbs distributions, with local temperatures given by the ordered Dirichlet distribution [3]. As a consequence, we prove that the empirical energy profile satisfies a large deviation principle, with a rate function in agreement with the prediction of Macroscopic Fluctuation Theory [4]. If time allows, we shall discuss ongoing work (in collaboration with T.Sasamoto) about dynamical large deviations.

[1] R. Frassek, C. Giardinà, J. Kurchan, "Non-compact quantum spin chains as integrable stochastic particle processes", J. Stat. Phys. 180, 135-171 (2020) 
[2] R. Frassek, C. Giardinà, "Exact solution of an integrable non-equilibrium particle system", J. Math. Phys. 63, 103301 (2022)
[3] G. Carinci, C. Franceschini, D. Gabrielli, C. Giardinà, D. Tsagkarogiannis, "Solvable stationary non-equilibrium states", J. Stat. Phys. 191, 10 (2024)
[4] G. Carinci, C. Franceschini, R. Frassek, C. Giardinà, F. Redig, "Large deviations and additivity principle for the open harmonic process" Comm. Math. Phys. 406, 103 (2025)

Periodic PushASEP model is a bidirection interacting particle system with N particles moving on a torus of size L. This is a generalization of the unidirectional TASEP model. We are interested in the behavior of the system when N, L→∞ with N/L→ρ at the specific time-scale of t ~ L^{3/2}. For the flat and step initial conditions, we are able to compute the limiting one-point distributions, which are interpolations between a normal distribution and a Tracy-Widom distribution. These limiting distributions first appeared in [Baik and Liu, 2018]. Based on joint work with Axel Saenz (Oregon).

Abstract: I will show how one can use Riemann-Hilbert techniques to study domino tilings of L-shaped Aztec domains, which consist of Aztec diamonds with one or several corners removed. In particular, I will describe a transition which takes place when the removed corner becomes large. I will present detailed asymptotics for the number of domino tilings in this transition and an exact expression for the associated correlation kernel. Our next goal is to understand the asymptotics of the correlation kernel.
The talk will be based on joint work with Christophe Charlier and joint work in progress with Felix Gideonse.

Abstract: The Hopfield model stands as a paradigmatic neural-network model, acting as a crossroad where statistical mechanics, theoretical neuroscience, and machine learning merge.  While its foundation is biologically grounded, its formal equivalence with layered neural networks (such as Boltzmann machines) naturally links it to contemporary machine learning. Furthermore, due to its analytical tractability, the Hopfield model naturally provides an ideal testing benchmark for the development of theoretical tools aimed at understanding structured data and training processes, which are some of the current challenges in the field. In this talk, after a broad introduction to the model and its phenomenology, we will draw a picture of recent research directions, mainly using tools from statistical mechanics of disordered systems and random matrix theory.

Abstract:  I will discuss recent results on the probability of observing unusually large empty regions in a gas of noninteracting fermions. These rare fluctuations provide a sensitive probe of the correlations induced by the Pauli exclusion principle. In collaboration with the experimental group of Tarik Yefsah at the Laboratoire Kastler Brossel (LKB) in Paris, these effects can now be measured directly in two-dimensional quantum gases using quantum gas microscopy. As the temperature increases, the statistics evolve from strongly correlated quantum behavior to the uncorrelated Poisson limit. This crossover is described by a universal scaling function that displays an unexpected phase transition.

Abstract: We consider agents subject to multiple-wise interactions, i.e. when each particle interacts with m other ones at the same time. We derive the associated Vlasov equation and prove propagation of chaos. We then consider the hydrodynamic limit for monokinetic solution and derive the corresponding Euler equation. The precise estimates in N and m of the rate of convergence allows to consider the joint (conditional) limit of diverging N and m towards a new type of macroscopic equation involving a vector field derived out of the Hewit-Savage theorem and an unpublished result by Pierre-Louis Lions.

Abstract: In this talk I will present the Hastings-Levitov model in anisotropic regime, for which I will discuss the small particle limit. This model belongs to a broader class that simulate the formation of fractal structure in nature. Among the many models introduced for this purpose, the one considered here is defined through conformal maps which enable a rigorous analysis. Based on joint work with Vittoria Silvestri.

Abstract: I will discuss the integrability property of a stochastic and quantum deformation of the Rule 54 cellular automaton: the simplest microscopic (deterministic) reversible model in 1+1 discrete space and time dimensions with strong local interactions. First, I will introduce the Rule 54 model and its two deformations:

• In the stochastic case, I couple the system to stochastic boundary reservoirs and show that the resulting non-equilibrium steady states can be constructed explicitly in matrix product form.

• In the quantum case, I explain how the model can be embedded into the Yang-Baxter integrability framework and provide an expression of the Lax operator, the operator responsible for the existence of an extensive number of conserved quantities that constrain the dynamics. It turns out that Yang-Baxter integrability is more common than previously thought!

Based on 2603.25424 with T. Prosen.

Abstract:  The classical Heisenberg model in three dimensions displays a low temperature phase with broken continuous rotational symmetry, a fact originally proved by Froehlich, Simon and Spencer via Reflection Positivity and Infrared Bounds. Spin wave theory provides a way to systematically, although formally, calculate the low temperature corrections to the spontaneous magnetization, beyond the naive, lowest-order, value computed via the Gaussian approximation. In recent work, in collaboration with Sebastien Ott, we succeeded in proving the asymptotic nature of such a low-temperature spin-wave expansion. The proof combines a priori bounds on the moments of the local spin observables, following from Reflection Positivity, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. This generalizes an approach, proposed originally by Bricmont and collaborators in 1981 in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables. 

Abstract: We study the Ising model on a two-community stochastic block model, where $n$ spins are split into two equal groups with inter-community interaction parameter $\alpha_n\in[0,1]$. We provide a complete characterization of the phase diagram and show that, almost surely with respect to the graph realization, the model undergoes a uniqueness/non-uniqueness phase transition of the Gibbs measure. In particular, in the supercritical regime, the law of the magnetization vector of the two communities converges to a mixture of Dirac measures that, depending on whether $\alpha_n\gg 1/n$ or $\alpha_n\lesssim1/n$, is supported on two or four points, with possibly different weights. In the uniqueness region, we further analyze the fluctuations of the magnetization vector in the subcritical regime and we prove a quenched central limit theorem.

Abstract: The Kipnis, Marchioro and Presutti process (KMP), introduced in the 80’s in [1], is a simple Markovian model for energy exchange in a chain of harmonic oscillators. In this talk I aim to shed light on the connection of this process to the Kardar-Parisi-Zhang (KPZ) equation, under appropriate scaling. This is a joint work with Guillaume Barraquand [2].

[1] Kipnis Claude, Carlo Marchioro, and Errico Presutti. ”Heat flow in an exactly solvable model.” Journal of Statistical Physics 27 (1982): 65-74.
[2] Barraquand Guillaume, Casini Francesco, ”Convergence of the KMP model to the KPZ equation”, ArXiv 2507.19222.

Abstract: The model I will consider is a version of the Polynuclear Growth Model (PNG). PNG describes unit-speed nucleations emerging from randomly positioned seeds in space-time. Upon meeting each other, these nucleations cease to grow, but other nucleations will still emerge from further space-time seeds. The model is also closely connected to the problem of longest increasing subsequences in random permutations. PNG is a prominent member of the KPZ universality class of growth models, named after the famous Kardar-Parisi-Zhang equation. Models in this class feature strong interactions that make them difficult to handle, and alter the classical 1/2-power scaling and Gaussian limit laws to 1/3-power scalings and limit distributions known from random matrix theory.
The t-PNG variant I will consider flips a coin every time nucleations meet, and only stops these growing with probability 1-t. With the remaining t probability a new nucleation is started instead, and this makes the analysis of the model more difficult than classical PNG. I will explain a stationary scenario for this version due to Drillick and Lin, and describe our result which is local convergence to this stationarity. Some non-trivial coupling arguments were necessary, which I will explain in the talk.

Abstract: The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this talk, I will revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble.

Abstract: Complex systems tend to exhibit out-of-equilibrium dynamics over a broad range of timescales. A key theory challenge is to understand the features of this out-of-equilibrium behavior from the properties of the attractors of the system’s dynamical equations. Mean-field theories of spin glass dynamics offer elegant examples, linking phenomena such as aging to the properties of special families of stationary points of the underlying free-energy landscape, that attract the dynamics at large times. However, these insights apply mainly to systems to which one can associate such landscapes. It is an open question how to extend these ideas to high-dimensional non-conservative systems (such as biological neural networks, large ecosystems) whose dynamics is not landscape optimization. I will present a simple model of a high-dimensional system with non-reciprocal interactions whose out-of-equilibrium, chaotic dynamics can be characterized analytically at long times. I will discuss its dynamical phase diagram and compare it to the statistical distribution of the many, unstable fixed points of the dynamical equations. This comparison challenges the idea that chaotic dynamics in non-conservative settings can be understood from fixed points alone, at least from the typical ones. The results are obtained through a combination of the Kac-Rice formalism, random matrix theory and dynamical mean-field theory.

Abstract: As any discrete dynamics in classical simplex  $\Delta_N$ can be described by a stochastic matrix,  $p'=Sp$, we introduce an analogous dynamics in the space of  quantum measurements, often referred to as positive operator-valued measurement (POVM). Such a measurement is defined by set of positive operators, $P_j = P_j^{\dagger}\ge 0$, summing to identity. Transformations in the set of quantum measurements can be described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices.  Such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices, sometimes called quantum magic squares. Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in the set of quantum measurements.
* joint work with Albert Rico (Siegen), Moises Moran (Hong-Kong) and Shmuel Fridland (Chicago)

Abstract: We study branching random walks in spatial random environment on the integers. For a fixed offspring distribution μ and given a realisation of the environment ω, at each time step every particle produces offspring (independently of all other particles and of the environment) according to into μ. Each offspring born from a parent located at i moves to i +1 with probability ω_i and to i -1 with probability 1 - ω_i . We prove a CLT for the location of the rightmost particle. We also discuss universality of the behaviour of the transition probabilities.
Joint work with Alice Callegaro and Carlo Scali.

Abstract: In this talk, we will examine the role of determinantal processes in modeling the repulsion among points. We will discuss the connections between these stochastic processes and quantum mechanics, through the measurement of Slater determinant states, that reflect fermionic behavior. The goal is to establish novel quantitative bounds between these objects, exploiting these connections and measures such as trace/total variation and Wasserstein distances. Based on a joint work with C. Boccato and F. Pieroni, Ann. Henri Poincaré (2026). https://doi.org/10.1007/s00023-026-01658-3.

Abstract: We show that powers of characteristic polynomials of matrices with independent, identically distributed entries converge to the Gaussian Multiplicative Chaos measure. This result holds both for real and complex matrices and for general entry distribution.

Abstract: We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural positivity hypotheses on the associated transfer operators, we prove the existence of thermodynamic limits for expectations of local observables and establish almost sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially with the distance between sites, with a deterministic rate. In the i.i.d. case, exponential decay again holds with a deterministic rate, and the probability of observing the predicted decay approaches one exponentially fast in the separation. For strictly stationary ensembles with decaying spatial dependence, correlation decay quantitatively reflects the mixing profile: $\rho$--mixing yields polynomial bounds with high probability, while stretched-exponential (resp.\ exponential) decay in $\rho$ (resp.\ $\beta$) yields stretched-exponential (resp.\ exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic and Gaussian translation-invariant ensembles, and provides a random transfer-operator route to typical correlation decay.

Abstract:  Weak convergence of the centered and scaled sequences of largest eigenvalues of $n \times n$ GUE matrices, and of the lengths of the longest increasing subsequences of uniformly random permutations of length $n$, to the GUE Tracy-Widom distribution, are two primary examples of the KPZ fluctuations. In this talk we address a question posed by Kalai: what is the analogue of the classical law of iterated logarithm for these sequences? In particular, we show that after a further scaling of $(\log n)^{2/3}$ (resp.\ $(\log n)^{1/3}$), the $\limsup$ (resp.\ $\liminf$) of the centered and scaled sequences almost surely converge to some non-zero and finite constants, which we explicitly determine. This answers a question left open by Paquette and Zeitouni. One key ingredient in solving these problems was understanding the scale at which decorrelation occurs in these sequences. For this we use the connection with the Brownian and Poissionan last passage percolation models and use the geometric inputs from these models.
This is based on a joint work with Baslingker, Basu, and Krishnapur (arXiv:2410.11836).

Abstract:  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.

Abstract:  We study a natural modification of first-passage percolation. Consider a graph G with a reference vertex o. Place a red particle at o and colorless particles, which we call seeds, everywhere else. The process spreads at rate 1 and starts at o, while seeds are dormant. Seeds reached by the process turn red and keep spreading first-passage percolation (at rate 1). All vertices have independent exponential clocks which ring at rate γ > 0; when a clock rings, the corresponding red vertex turns black. We establish asymptotic (in time) upper and lower bounds on the size of the longest red path and largest red cluster when G is a tree, deterministic or random. (Joint work with Tom Garcia-Sanchez.)

Abstract: The node2vec random walk is a non-markovian random walk on the vertex set of a graph used in various applications for network embedding. It is defined in terms of 3 tuning parameters to penalize or reward back-tracking moves and moves to neighboring vertices belonging or not to local triangles. From a mathematical perspective, it is a second-order Markov chains generalizing in a non-trivial way the non-backtracking random walk. I will discuss how its basic fundamental structure on arbitrary graphs emerge when we lifted it to the spaces of directed edges and wedges. Such liftings allow in fact for different useful Markov representations out of which we can determine ergodic, reversible and recurrent properties in terms of the underlying (finite or infinite) graph structure. As we shall see, unlike the non-back tracking companion which has a special general bistochastic nature, node2vec is intrinsically harder unless the underlying graph is d-regular.