Titles and Abstracts of Talks
One-sided versus two-sided continuity. Long-range aspects, two examples and the role of entropic repulsion.
A.C.D. van Enter (Groningen)
We compare g-measures, aka Doeblin measures, which have one-sided continuous conditional probabilities with Gibbs measures which have two-sided continuous conditional probabilities. We show by discussing two examples, namely low-temperature Gibbs measures for Dyson models, and the Schonmann projection, the one-dimensional marginal of the two-dimensional Ising model, that neither property implies the other one.
Both examples can be understood with help of the concept of entropic repulsion.
Joint work with R. Bissacot, E. Endo and A. Le Ny, and with S.B. Shlosman.
Also, if some time is left, some further roles which the long-range parameter in the Dyson model has played are discussed.
Annealed Potts Models on Rank-1 Inhomogeneous Random Graphs
Claudio Giberti (Modena)
In this talk, we present recent results on the annealed ferromagnetic q-state Potts model on sparse inhomogeneous random graphs of rank-1 type, a widely used framework for modeling complex networks with heterogeneous connectivity. Within the annealed setting, where both spins and graph randomness are averaged on equal footing, we rigorously map the model to an inhomogeneous Curie–Weiss system and derive a variational representation for the thermodynamic pressure.
We establish the existence of the thermodynamic limit under mild assumptions on the vertex weights and provide a detailed analysis of the phase diagram. A key finding is that the nature of the phase transition strongly depends on the moments of the weight distribution. In particular, when the weight variance is infinite, the system exhibits spontaneous magnetization at any positive temperature, implying the absence of a phase transition. In contrast, for finite-variance weights and q ≥ 3, we show that the phase transition is generically first order, characterized by a discontinuous jump in the order parameter, and that this behavior persists under small external fields.
For networks with Pareto-distributed weights, we uncover a transition from first- to second-order behavior depending on the power-law exponent, highlighting a smoothing phenomenon driven by network heterogeneity. These results provide a comprehensive and rigorous picture of phase transitions in annealed Potts models on complex networks.
[Joint work with Cristian Giardinà, Remco van der Hofstad, Guido Janssen, and Neeladri Maitra.]
Large-scale dynamics in aggregation models with product kernels.
Stefan Grosskinsky (Augsburg)
We study mean-field limits of interacting particle systems with interaction kernels of product form. One example is the inclusion process with a bilinear kernel, where the partition of total mass evolves according to the well-studied Poisson-Dirichlet diffusion in the scaling limit. This complete result relies on self-duality properties of the model, which are not present for generalized models with a non-linear product kernel such as the exchange-driven growth process introduced in [PRE 68, 031104 (2003)]. Depending on the non-linearity, the system may exhibit (instantaneous) gelation and our results for the scaling limit are only partially rigorous so far.
This is joint work with Paul Chleboun, Simon Gabriel and Angeliki Koutsimpela.
Some aspects of the scaling limit of the 2D Abelian sandpile
Antal Járai (Bath)
In the Abelian sandpile model, particles are located on the vertices of a finite connected graph, with one of the vertices playing the role of a “sink” that carries no particle. Whenever the number of particles on a vertex is at least the degree, the vertex “topples” and sends one particle to each neighbour, with particles arriving at the sink lost. The system evolves in discrete time, by adding a particle at a random vertex and “stabilizing” via carrying out all possible topplings. In this talk, I will consider the stationary distribution of the Abelian sandpile, where the graph is a lattice approximation of a region U lying in the complex plane. In the first part (joint work with M.W. Elvidge), I will discuss sufficient conditions on the lattice that ensure that the scaling limit of the height-zero probability is conformally covariant, generalizing a result obtained by Adame-Carillo and Ruszel on the square lattice as part of their investigation of the fermionic DGFF. In the second part, I will discuss two results on sandpile waves started at a point o in U: (i) a limit theorem on the radius of a typical wave; and (ii) the scaling of the probability that a typical wave conditioned on containing a point z in U (different from o) comes within a small distance of the boundary.
Metastates, overlaps, and continuous symmetry breaking in the random field spherical model
Christof Külske (Bochum)
We study the large-volume asymptotics of the ferromagnetic spherical model for d-dimensional local spins, in the presence of quenched vector-valued random fields. At low temperature the model shows continuous symmetry breaking, where the ordering is spontaneously broken by the random-field fluctuations, and chaotic size dependence occurs. We give a description in terms of limiting theorems for metastates and overlaps. We also discuss the effect of random fields in different volume scalings, for which some spin-glass-type characteristics may appear, with transitions depending on the scaling regimes.
Joint work with Kalle Koskinen, GSSI, L’Aquila.
arXiv:2505.16843
Spherical and unitary designs from a physicist point of view
Jorge Kurchan (Paris)
Designs are sets of points chosen so that integrating polynomials up to degree t on the whole space (sphere, unitaries…) gives the same result as summing over the set of points. This turns out to be the stat-mech of a set of particles with a repulsive potential, and has the phenomenology expected for such a system. Interestingly, the difficulty comes from the demand that configurations have coordinates within a field, which makes the problem immensely more difficult.
Roughening transitions for long-range Discrete Gaussian Chains
Arnaud Le Ny (Paris)
Discrete Gaussian Chains, also called Integer-Valued Gaussian Free Field (IV-GFF) as it coincides with GFF conditoned to integer values, have been shown to exhibit localisation/delocalisation roughening transitions depending on dimensions, temperatures, interactions, etc. In this talk, we shall describe recent progress on qualitative statements and provide sharp quantitative estimates in one dimension for long-range DGC depending also on the decay of the polynomial interaction, as very long-ranges allows to recover the roughening transition famous in dimension two for nearest neighbours models.
Joint works with Christopher Chalhoub (London), Loren Coquille (Grenoble), Paul Dario (Cergy), Aernout van Enter (Groningen), Corentin Faipeur (Cergy), Wioletta Ruszel (Utrecht).
References:
[1] L. Coquille, A. van Enter, ALN, W. Ruszel. Qualitative Delocalisation of long-range Discrete Gaussian Chains. Journal of Statistical Physics 2024.
[2] L. Coquille, P. Dario, ALN. Quantitative delocalisation for the Gaussian and q-SoS long-range chains. Annales Henri Lebesgue 2026.
[3] C. Chalhoub, P. Dario, C. Faipeur, ALN. Scaling limits and quantitative delocalisations for disordered long-range Dicrete Gaussian Chains. Preprint 2026.
Relaxation to Nonequilibrium
Christian Maes (Leuven)
We describe the structure of relaxation for a steadily driven macroscopic body. The time-evolution is characterized as the zero-cost flow for a nonequilibrium and nonlinear extension of the Onsager-Machlup action governing the dynamical fluctuations. The approach hinges on two main elements: the principle of local detailed balance, which identifies the relevant thermodynamic forces, and the canonical decomposition of the frenesy into a Legendre pair. Notably, it is the time-symmetric component of the Lagrangian, the frenesy, that shapes the structure of the macroscopic evolution for given forcing. The nonequilibrium entropy, which governs the static macroscopic fluctuations of the system, is monotone in time. The results can be interpreted as the steady nonequilibrium extension of GENERIC where relaxation to equilibrium is governed by a dissipative gradient flow superimposed on a Hamiltonian flow.
Ref: CM and Karel Netočný, Relaxation to nonequilibrium. Journal of Non-Equilibrium Thermodynamics. https://doi.org/10.1515/jnet-2026-0037
Binary Structures on Infinite-dimensional Statistical Manifolds
Jan Naudts (Antwerpen)
Information geometry introduces the notion of a dually flat geometry. The duality referred to is that of Thermodynamics. The dual parameters of a parameterized statistical model are defined by Legendre transformation. An exponential family of models in Statistics corresponds with a Gibbs state in Statistical Mechanics. In 1979 Ruppeiner linked curvature of the Levi-Civita geometry to critical behavior and phase transitions. Information Geometry emphasizes that the natural geometry is not that of Levi-Civita but is a flat geometry, with affine parameters and vanishing connection coefficients Γkij. The dual parameters are affine parameters for the dual geometry. An important role is attributed to divergences. They are called relative entropies in Statistical Mechanics.
A statistical manifold is a manifold of non-degenerate probability measures. The tangent spaces are signed measures. Pistone and coworkers started in 1995 to study infinite-dimensional statistical manifolds. They showed that with an appropriate norm the tangent spaces become Banach spaces. The space of probability measures absolutely continuous w.r.t. the reference measure becomes a Banach manifold.
For practical reasons most statistical models have a finite number of parameters n and can be described as a sub-manifold of the Euclidean space Rn. However, it is more natural to require that the statistical model is a sub-manifold of an infinite-dimensional Banach manifold of probability measures.
In Statistical Mechanics a relevant manifold is the phase space. It has two types of tangent vectors: a change in position and a change in momentum.
The quantum state space is the manifold of non-degenerate density operators on a Hilbert space. It is the analogue of the space of non-degenerate probability measures on the classical phase space. The two types of observables are less visible but are present. They determine the complex structure of Quantum Mechanics. After complexification the two types of observables are the self-adjoint, respectively anti-self-adjoint operators.
References
[1] J. Naudts and J. Zhang, Legendre duality: from thermodynamics to information geometry, Info. Geo. 7, 623–649 (2024).
[2] J. Naudts, A Complex Structure for Two-typed Tangent Spaces, Entropy 27, 117 (2025).
[3] J. Naudts, Binary Structures on Banach Spaces, Axioms 15, 300 (2026).
Hydrodynamic limit of the directed exclusion process
Ellen Saada (Paris)
In a joint work with Assaf Shapira and Federico Sau, we derive the Euler (hyperbolic) hydrodynamic limit for the directed exclusion process (DEP), a one-dimensional conservative interacting particle system that preserves particle–hole symmetry while breaking left–right symmetry. The proof relies on an explicit multi-process coupling which guarantees a strong form of attractiveness and macroscopic stability for the particle system.
Large deviations for interacting particle systems with large spin
Tomohiro Sasamoto (Tokyo)
Large deviation theory has been a central subject in the study of interacting particle systems. Since the seminal work by Kipnis, Olla, and Varadhan (1989) on the symmetric simple exclusion process (SEP) [1] and the subsequent introduction of macroscopic fluctuation theory (MFT) by L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim [2], a wealth of results has been accumulated. Nevertheless, various outstanding issues remain.
Recently we have proposed a new framework of large deviations for a class of interacting particle systems which have a parameter “s” called a spin. The class contains many interesting particle systems such as the partial exclusion process, inclusion process, harmonic process etc, many of which were introduced by Frank Redig and a few coauthors [3]. For this class of models, we consider their large deviations for large s, while keeping the lattice. The scheme can be regarded as a lattice analogue of KOV and MFT.
In this talk, we present the large deviation principle (LDP) established for several models and showcase an exact formula for the rate function of the integrated current for the harmonic process. We will review these recent developments and discuss remaining challenges and future directions. This talk is based on collaborations with Cristian Giardina, Hayate Suda, and Kirone Mallick [4].
References:
[1] C. Kipnis, S. Olla, S.R.S. Varadhan, Hydrodynamics and large deviations for simple exclusion processes, Comm. Pure Appl. Math., 42 (1989), 115-137.
[2] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory, Rev. Mod. Phys., 87 (2015), 593-636.
[3] C. Giardina, J. Kurchan, F. Redig, K. Vafayi, Duality and Hidden Symmetries in Interacting Particle Systems, J. Stat. Phys. 135 (2009), 25-55; C. Giardina, J. Kurchan, F. Redig, Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes, J. Stat. Phys. 180 (2020), 135-171.
[4] C. Giardina, T. Sasamoto, Large spin large deviations for interacting particle systems; C. Giardina, T. Sasamoto, Path space large deviation on a lattice for interacting particle systems; C. Giardina, K. Mallick, T. Sasamoto, H. Suda, Exact solution of discrete macroscopic fluctuation theory for an integrable spin system, in preparation.