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# Mini Workshop on Reinforcement and Statistical Mechanics

Forschung, Probability Theory | | Workshop, Event

The aim of the workshop is to bring together people interested in statistical mechanics and in particular processes with reinforcement, both from the analytic and probabilistic point of view. There will be three minicourses for non-specialists. Everybody who is interested is welcome to participate. People associated to any project of the DFG priority program SPP 2265 Random Geometric Systems Pfeil are welcome to apply for funding by sending an email to one of the organizers.

## Organizers

Margherita Disertori (Bonn), Franz Merkl  (München), and Silke Rolles  (München)

The Mini Workshop takes place on Thursday, November 10, and Friday, November 1, 2022 at the Technical University of Munich in Garching. It is funded by the DFG priority program SPP 2265 Random Geometric Systems within the project Emergence of macroscopic phenomena in the non-linear hyperbolic supersymmetric sigma model.

## Speakers

There wil be minicourses on the following topics:

• Introduction to the renormalization group by Roland Bauerschmidt (Cambridge): Phase transition for the Arboreal Gas in $$d\geq 3$$
Abstract: Building on the previous talk by T. Helmuth, I will explain the main ideas of our proof with N. Crawford that the Arboreal Gas has a phase transition in dimension $$d\geq 3$$. This proof uses a renormalisation group approach in combination with Ward identities. As a result, we obtain the existence of a macroscopic component on finite tori, as well as free field like fluctuations for connection probabilities resulting from the spontaneously broken continuous symmetry ofthe nonlinear sigma model to which the Arboreal Gas is exactly related.
• Introduction to multiscale analysis by Constanza Rojas-Molina (Cergy-Pontoise): The Multiscale Analysis Method in the Theory of Random Schrödinger Operators
Abstract: We give an overview of the Multiscale Analysis Method (MS), aversatile tool used to prove the existenceof pure point spectrum with exponentially bound states in random Schrödinger operators. Initially developed by Fröhlich and Martinelli in the early 8os, this technique has been the subject of active research through theyears, resulting in a refined method that applies to a wide variety of random models. The MSA allows to show the absence ofquantum transport thus giving a rigorous proof of P.W. Anderson's observations on dynamical localization in disordered systems.
• Sabine Jansen (München): Witten Laplacian for spin systems vs. Stein's method in probability theory
Abstract: Helffer and Sjöstrand pioneered analytic techniques to prove decayof correlations for Gibbs measures for real-valued spins on a lattice. The method involves differential operators related to deformed Laplacians introduced by Witten. Stein's method was originally introduced to quantify errors in the normal approximation; central limit theorems areobtained as a byproduct. The talk presents some key ideas from both methods and highlights some surprising similarities.

There will be the following research talks:

• Tyler Helmuth (Durham): The Arboreal Gas
Abstract: In Bernoulli bond percolation, each edge of a graph is declared open with probability p, and closed otherwise. Typically one asks questions about the geometry of the random subgraph of open edges. The arboreal gas is the probabilitymeasure obtained by conditioning on the event that the percolation subgraph si a forest, i.e., contains no cycles. Physically, this is a model for studying the gelation of branched polymers. What are the percolative properties of these random forests? Iwill discuss some of what is known and conjectured, and will highlight the important role played by a spin model representation of the arboreal gas.
Based on joint workswith R. Bauerschmidt, N. Crawford, and A. Swan.

• Rémy Poudevigne (Cambridge): Slow phase transition for the VRJP on the tree
Abstract: Joint work with Peter Wildemann (Cambridge). The vertex reinforced jump process (VRJP) is closely linked to the edge-reinforced random walk, one of the simplest and oldest example of areinforced random walk. On $$Z^d$$ it is known that the VRJP goes from arecurrent phase to a transient phase but the behaviour around the critical point is not well understood. Here we look at the behaviour of the VRJP around the critical point on the d-ary tree (in the transient phase) for some quantities like the time spent at the origin and compute the speed of the phase transition. This result is based on a link between the VRJP on trees and branching random walk.

• Christophe Sabot (Lyons): Stochastic calculus aspects of the Vertex Reinforced Jump Process
Abstract: In thistalk we will present somerelations between classical computations on exponential functional of the Brownian motion and some representation of the Vertex Reinforced Jump Process (VRJP). In particular we will show how the Lamperti transformation, the Markov property of the Matsumoto Yor process have natural generalizations to the multidimensional setting involvingthe random potential associated with the VRJP.

• Xiaolin Zeng (Strasbourg): The random Schrödinger operator related ot $$H^{2|2}$$ model, integrated density of state
Abstract: We recall the definition of the random Schrödinger operator related to the supersymmetric hyperbolic sigma model, and we discuss some results about its spectrum, in particular, the Wegner estimate and the behavior of IDOS at the bottom of the spectrum.
Based on joint works with Margherita Disertori and Constanza Rajas-Molina.

## Program

Thursday, November 10, 2022

• 13:30-14:20 Research talk by Tyler Helmuth
• 14:30-16:00 Minicourse by Roland Bauerschmidt
• 16:00-16:30 Coffee break
• 16:30-17:20 Research talk by Rémy Poudevigne

Friday, November 11, 2022

• 08:30-10:00 Minicourse by Constanza Rojas-Molina
• 10:00-10:30 Coffee break
• 10:30-11:20 Researchtalk by Xiaolin Zeng
• 11:30-12:20 Research talk by Christophe Sabot
• 12:20-14:00 Lunch break
• 14:00-15:30 Minicourse by Sabine Jansen

The talks take place ni room El 02.5901.013 on the second floor of the building for Electrical and Computer Engineering, Hans-Piloty-Straße 1, 85748 Garching. To get there by public transport take the subway U6 to the final destination Garching-Forschungszentrum.
All talks wil be live streamed via zoom. If you would like to participate via zoom, please send an email to Silvia Schulz Silvia.Schulz@ma.tum.de for the zoom link.