The aim of the workshop is to bring together people interested in random interacting systems. Everybody who is interested is welcome to participate.

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Titles and abstracts

**Rob van den Berg:** An OSSS-type inequality for uniformly drawn subsets of fixed size.

*Abstract: *The OSSS inequality (O'Donnell, Saks, Schramm and Servedio, 2005) gives an upper bound for the variance of a function of independent 0-1 valued random variables in terms of the influences of these random variables and the computational complexity of a (randomised) algorithm for determining the value of the function.

Duminil-Copin, Raoufi and Tassion, 2019, obtained a generalization of the OSSS inequality to monotonic measures and used it to prove new results for Potts models and random-cluster models. That generalization naturally triggers the question if there are still other measures for which such an inequality holds.

This talk concerns an OSSS-type inequality for a family of measures that are clearly not monotonic, namely the k-out-of-n measures (these measures correspond with drawing k elements from a set of size n uniformly).

The talk is related to my paper https://arxiv.org/pdf/2210.16100.pdf I also plan to make some (very) brief remarks concerning work in progress with Henk Don.

**Jiří Černý:** On the concentration of the complexity of spin-glass Hamiltonians.

*Abstract:* In the first part of the talk, I review some results on the complexity and topology of the energy landscapes of spherical spin glasses, which can be proved with help of Kac-Rice formula, using careful random matrix analysis. These energy landscapes can be viewed as random polynomials on a high-dimensional sphere. The distribution of their critical points and topology should play an important role in rigorous understanding of the dynamics of spin glasses. In the second part, I present new concentration results for these objects. The talk is based on a work in preparation, jointly with D. Belius.

**Nick Crawford:** The statistical mechanics of forests and fermionic sigma-models.

**Antal Jarai:** A survey of sandpile models and some recent results.

*Abstract: *In the first half of this talk I will give an introduction to sandpile models in the context of self-organized criticality, accessible to a general audience in stochastics. While not intended as a model of real sand or other granular media, the dynamics of sandpile particle systems mimic important features of real avalanches. A key property of interest is that adding a single particle to the system triggers a response characterized by heavy tailed distributions. In the second half of the talk I will report on recent progress on the scaling limit of waves in the 2D model and discuss several open problems.

**Achim Klenke:** Biased Random Walk on Spanning Trees of the Ladder Graph. Abstract

**Roman Kotecky:** Surface tension for the Widom-Rowlinson model

*Abstract:* The topic is a requisite constituent in a series of papers under preparation, jointly with Frank den Hollander (Leiden), Sabine Jansen (Munich), and Elena Pulvirenti (Delft). While our main topic is metastability for the Widom-Rowlinson model of interacting particles in continuum, here I will concentrate on a well separated part devoted to microscopic description of the surface tension at the "liquid/gas" coexistence line.

I will discuss how surface layer fluctuations lead to a derivation of the low temperature asymptotics of the surface tension with an entropic term featuring fractional power of the temperature. The analysis relies on a careful examination of the surface tension limit and its expression in terms of the spectral radius of an appropriately chosen transfer operator.

**Eva Kopfer:** Conformally Invariant Random Geometry on Manifolds of Even Dimension.

*Abstract: *We present a concise introduction to conformally invariant, log-correlated Gaussian random fields on compact Riemannian manifolds of general even dimension uniquely defined through its covariance kernel given as inverse of the Graham-Jenne-Mason-Sparling (GJMS) operator. The corresponding Gaussian Multiplicative Chaos is a generalization to the n-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly.

**Christian Mönch:** Inhomogeneous long-range percolation: recent results.

*Abstract: *The Weight-Dependent Random Connection Model combines long-range percolation in Euclidean space with scale-free network models. The model and its variants have received considerable attention in the last few years. The talk aims to provide an overview of recent results, where special consideration will be given to the "weak decay regime". Here, connection probability tails are heavy enough to circumvent many geometrical difficulties that arise in short-range percolation models in low dimensions. I will summarise known sufficient conditions for existence and transience of an infinite cluster and discuss a local existence theorem which improves upon a result of Berger (2002) for homogeneous long-range percolation. The theorem is the main tool to establish continuity of the percolation function throughout the weak decay regime in dimension at least two. If time permits, I will discuss some open questions and future research goals. The talk is partly based on joint works with P. Gracar, L. Lüchtrath, M. Heydenreich and P. Mörters.

**Balazs Rath:** The window process of slightly subcritical frozen percolation.

*Abstract:* The mean field frozen percolation process is a dynamic random graph model which starts with the empty graph on \(N\) vertices, an edge between a pair of vertices is added at rate \(\frac{1}{N}\) and connected components of size \(k\) are deleted at rate \(r \cdot k\), where\( r = r(N)\) is a constant that depends on \(N\). This model is known to exhibit self-organized criticality when \(1 \ll N\) and \(\frac{1}{N}\ll r(N) \ll 1\), i.e., the dynamics keep the graph in a state which is essentially a near-critical critical Erdős-Rényi graph. One defines the window process \(w(t) = A(t) \cdot \frac{t}{N}\), where \(A(t)\) is the number of vertices alive at time \(t\). We derive scaling limits for the time evolution of \(w(t)\) when \(r(N) = n^a\) for some \(-\frac{1}{3} < a < 0\), thus giving a detailed picture of the mechanism that produces the self-organized criticality of the model. Joint work with Márton Szőke (BME) and Dominic Yeo (King's College London).

**Artem Sapozhnikov:** Uniqueness of the infinite connected component for the vacant set of Brownian interlacements.

*Abstract:* Brownian interlacements is a Poisson cloud of doubly-infinite Wiener sausages of fixed radius in Euclidean space, which is a continuum analogue of Sznitman's random interlacements. Points of Euclidean space not covered by any of the sausages form the vacant set of Brownian interlacements. In this talk, I discuss the proof of uniqueness of the infinite connected component in the vacant set of Brownian interlacements. One of the main ingredients in the proof is a sharp estimate on the probability that the vacant set of a finite ensemble of independent Wiener sausages intersects a given tiny ball in at least two connected components. Joint work with Yingxin Mu.

**Sasha Sodin:** The eigenfunctions of tridiagonal \( G_\beta E \) matrices

*Abstract:* We shall discuss the shape of the eigenfunctions of the Dumitriu–Edelman tridiagonal random matrices the eigenvalues of which are distributed according to the law of the \(\beta\)-gas with Gaussian potential. The problem in infinite volume has been studied by Breuer—Forrester—Smilansky, whereas we investigate the behaviour in finite volume; it turns out that the answers to these two questions are completely different. Based on joint work in progress with Ofer Zeitouni.

**Jeff Steif:** Poisson Representable Processes.

*Abstract:* Motivated by Alain-Sol Sznitman's random interlacement process, we consider a general class of processes which can be constructed in a similar manner. Namely, one considers a general Poisson process on the collection of subsets of a given set S and the corresponding "Poisson Generated Process" is obtained by taking the union of the sets which arise in the Poisson process. In this way, we obtain a random subset of S or equivalently a 0-1 valued process indexed by S. Our main focus is to determine which processes are representable in this way. Some of our results are as follows. (1) All positively associated Markov chains and a large class of positively associated renewal processes are representable. (2) Whether an average of two product measures, with close densities, on n variables, is representable is related to the zeroes of the polylogarithm functions. (3) Using (2), we show that a number of tree indexed Markov chains as well as the Ising model on \({\mathbb Z}^d\) for d at least 2 for certain interaction parameters, is not representable. (4) The collection of permutation invariant processes which are representable corresponds exactly to the set of infinitely divisible random variables on \([0,\infty]\) via a certain transformation. (5) The supercritical Curie-Weiss model is not representable for large n. The talk is based on joint work with Malin P. Forsström and Nina Gantert.

**Theo Sturm:** Wasserstein Diffusion on Multidimensional Spaces *Abstract*

**Andrew Swan:** A new hyperbolic sigma model. *Abstract.*

**Peter Wildemann: **Reinforced Random Walk and a Supersymmetric Spin System on the Tree

*Abstract:*** **Motivated by predictions about the Anderson transition, we study two distinct but related models on regular tree graphs: The vertex-reinforced jump process (VRJP), a random walk preferring to jump to previously visited sites, and the \(H^{2|2}\)-model, a lattice spin system whose spins take values in a supersymmetric extension of the hyperbolic plane. Both models undergo a phase transition, and our work provides detailed information about the supercritical phase up to the critical point: We show that their order parameter has an essential singularity as one approaches the critical point, in contrast to algebraic divergences typically expected in statistical mechanics. Moreover, we locate an additional multifractal intermediate phase on large finite trees. This talk is based on arxiv:2309.01221 and is joint work with Rémy Poudevigne.

## Location and dates

**Tuesday 27**^{th} Februray to Thursday 29^{th} 2024, at the Department of Mathematics, Technische Universität München.

All talks take place in room 5604.EG.011 (Hörsaal 2, Boltzmannstr. 3, 85748 Garching b. München). There is also a transmission via zoom. Anybody who wants to participate virtually is invited to send an email to Nicole Maier at nj.maier@tum.de in order to obtain the zoom link.