## Participants :

Amine Asselah (organiser), Quentin Berger (organiser), Clément Cosco, Alex Drewitz, Nina Holden, Yueyun Hu, Tom Hutchcroft, Hubert Lacoin, Gregory Lawler, Julien Poisat, Aran Raoufi, Bruno Schapira, Vincent Tassion, Niccolo Torri, Yvan Velenik

by Amine Asselah and Quentin Berger

*25 – 29 March, 2019*

## Abstract

The workshop was devoted to Probability Theory, and focused on two celebrated models: random walks and percolation. 15 mathematicians, juniors and seniors, gathered for 5 days and shared two mini-courses, 7 lectures, and one open-problem session. Enough time was left for discussions, and undoubtedly many new collaborations will spring out of Les Treilles’workshop.

**Key words : **probability, random walks, percolation, scaling limits

## Review

The workshop gathered 15 mathematicians specialists of Probability Theory. It was organized around two mini-courses, each of 4 hours long: *Loop-erased random walks* by **Gregory Lawler** of Chicago University and *Scaling exponents in spanning forests* by **Tom Hutchcroft** from Cambridge University. There were seven talks, each lasting an hour, by leading scientists from a balanced group of juniors and seniors: Alex Drewitz (Köln), Clément Cosco (Paris), Nina Holden (Zurich), Hubert Lacoin (Rio), Aran Raoufi (Zurich), Vincent Tassion (Zurich) and Yvan Vélénik (Geneva). Time was also allocated for scientific discussions (inspired by the mini-courses or the different talks), and to an open problem session.

Two junior researchers just obtained Rollo Davidson price (Tom Hutchroft and Vincent Tassion), and Gregory Lawler received the Wolf price in January 2019. We had 5 juniors, 9 seniors and 1 Ph.D. student. All the participants were either specialists of percolation or random walks, two active subdomains of discrete probability theory.

The first course was presenting classical as well as very recent results on *loop-erased random walks (LERW)*, which is a variant of the simple random walk, a well studied random process made of a sum of independent increments on a d-dimensional lattice. The LERW uses the trajectory of a simple random walk but erases loops in chronological order. It produces an object with remarkable properties, much more tractable than the self-avoiding random walk, a useful model for polymer chemistry. First, relations between LERW, the loop soup and simple random walk were described. Then, intersection exponents were introduced for LERW. Dimension four was shown to be critical. Using slowly recurrent sets, Lawler obtained a description of LERW in dimension four. The existence of a two sided LERW was shown in dimension four. Then, the case of dimension two was studied in much details. Using conformal invariance, a limiting object can be obtained by scaling appropriately space and time. The course ended with Fomin’s identity, Schramm-Lowner evolution SLE_2, and culminated in the construction of two-sided LERW in dimensions two and three. Gregory Lawler invented loop-erased random walks in 1979-1980 as part of his Ph.D. thesis, and presented to us, some forty years later, a masterly and deep course focusing on the most salient features, and ending with perplexing open problems.

The second course was focusing on *uniform spanning forests*, which can be defined as subgraphs with no loops, spanning a given graph, such as the d-dimensional lattice. They have been introduced in 1847 by Kirchoff on the cubic lattice of Euclidean space, and the first lecture described the celebrated Wilson algorithm which uses loop-erased random walks (that Greg Lawler was covering) to build a random uniform spanning tree. Tom Hutchcroft presented a detailed analysis of spanning forests on a large class of graphs. In a recent paper, he used the theory of random interlacements of Sznitman to produce wired spanning forests, via the so-called Aldous-Broder algorithm. This new approach using random interlacements was developed, and shown to give sharp quantitative estimates for the diameter of trees. Along the way, some estimates on the Newtonian capacity of loop-erased random walk were used, and these estimates appear to be quite robust. They should also be useful in other contexts. Tom Hutchcroft is a junior faculty at Cambridge, and for his first research course, he chose beautiful ideas from two of his recent works, and presented them with utmost clarity.

The different talks presented recent advances in various domains of probability theory, in relation with statistical physics, random geometry or partial differential equations.

Yvan Velenik (professor at University of Geneva) and Aran Raoufi (postdoc at ETH Zürich) presented results on the Ising model, which is a discrete spin (+/-) model for the magnetization of a material, and has been widely studied since its introduction in 1920. **Yvan Velenik** showed how the interface between a ‘+’ and a ‘-‘ phase can be studied via Orstein-Zernike theory, and he described the scaling limit of this interface when tuning properly the parameters of the model. **Aran Raoufi**, on the other hand, focused on the correlations in the system, that is how the different spins influence each other. He showed us that except at some critical temperature, correlations decay exponentially with the distance, which can be interpreted as the fact that the influence of a spin is felt only up to a finite distance.

Vincent Tassion (junior professor at ETH Zürich) and Alexander Drewitz (junior professor at Köln University) presented their results on different percolation type models – they are used to describe porous media, and in particular how fluid can penetrate them. **Vincent Tassion** described renormalization techniques in the context of the standard percolation model, in which each bond is present or absent with a given probability p, independently of each other. He showed us how robust these techniques are, and showed us as an application that the critical parameter of the d-dimensional lattice can be approximated by that of the d-dimensional ‘slab’. **Alexander Drewitz** considered other types of percolation models (random interlacements and levels sets of the Gaussian Free Field), which present long-range dependence. He showed us how the two models he presented can be coupled to each other, and proved the existence of a non-trivial phase transition.

**Nina Holden** (postdoc at ETH Zürich) presented to us natural ways to parametrize random fractal curves in great generality. Random fractal curves appear naturally in many statistical mechanics models, and Nina Holden reviewed many examples where the so-called *natural parametrization*, via their so-called Minkowski content, has proven useful, in some of her most recent works : loop-erased random walks in dimension 2, percolation interfaces on random graphs, etc… These are deep results, in connection with many areas of probability theory.

**Clément Cosco** (Ph.D. Student at Paris Diderot University) described some of his results on the so-called KPZ equation, which has been the subject of an intense activity in recent years. The KPZ equation has been introduced to describe the random growth of an interface, and is related to many other statistical physics models. In particular, Clément Cosco described its relation to the directed polymer model, and showed to us that in dimension d≥3, the KPZ equation was tractable via some approximation procedure, at least in some ‘weak disorder’ regime.

**Hubert Lacoin** (professor at IMPA Rio de Janeiro) presented a model for a gas of charged particles in interaction. He explained its relation with the so-called Sine-Gordon model, and deduced that the model is well-defined in most cases. On the other hand, some problems with the definition of the model arise if the interaction between the particles decays only logarithmically with the distance : he showed to us how these difficulties can be interpreted, and how they can be overcome thanks to probabilistic techniques.

This week proved to be very stimulating for all participants. The mini-courses presented a rich material, the different talks were excellent, and the participants worked together in small groups during the time left between the lectures, which led to many mathematical progress. The two blackboards proved to be essential. The staff of Les Treilles created a kind atmosphere, and helped to make us feel at home in this beautiful domain that all of us adopted. We are confident that several fruitful collaborations will emerge from this workshop.