# Durham Symposium 2024

## Large-scale behaviour of critical and near critical statistical physics models

### 27-30 August 2024

The symposium will focus on the large-scale behaviour of critical and near-critical statistical physics models. Over the past two decades, significant breakthroughs have been made in understanding critical phenomena in such models. However, there is limited comprehension regarding near-critical phenomena, which lack the conformal invariances properties that characterise their critical counterparts. The symposium will explore this topic by focusing on recent developments in both critical and near-critical phenomena.

The event will take place over four days and will include ten full-length lectures given by prominent scholars in the field.

Participation is free, but registration is mandatory. For registration, please fill this form.

# Schedule

- All talks will take place in the Scott Logic Lecture Theatre (MCS001) at the Department of Mathematical Sciences
- Long format talks are 50 minutes long
- Short format talks are 20 minutes long

In both cases there should be 5 extra minutes for questions and discussion.

Tue |
Wed |
Thu |
Fri |
|||||||

09h30 |
Registration | 09h30 |
Sepúlveda | 09h30 |
Hartung | 09h30 |
Lambert | |||

09h45 |
Werner | 10h30 |
Break | 10h30 |
Break | 10h30 |
Break | |||

10h45 |
Break | 11h00 |
Pascal Lecture: | 11h00 |
Tough | 11h00 |
Jego | |||

11h15 |
Papon | Peltola | ||||||||

12h30 |
Lunch | 12h30 |
Lunch | 12h30 |
Lunch | |||||

14h00 |
Chhita | Free afternoon | 14h00 |
Cipriani | ||||||

15h00 |
Haunschmid-Sibitz | 15h00 |
Hegde | |||||||

16h00 |
Break | 15h30 |
Break | |||||||

16h30 |
Mahfouf | 16h00 |
Brummet | |||||||

17h00 |
Giles | 16h30 |
Schreuder | |||||||

17h00 |
Yuan | |||||||||

17h30 |
Drinks reception | 19h00 |
Conference dinner |

## Pascal Lecture

### Eveliina Peltola (Aalto University and Rheinische Friedrich-Wilhelms-Universität Bonn)

**Title: Loewner theory in complex analysis and random geometry**

**Abstract:**
About a hundred years ago, Charles Loewner introduced a recipe to encode planar growth into evolution of holomorphic maps. While his original motivation was purely in geometric function theory, this idea led to a success story also in the relatively modern field of random geometry, started about 75 years later. Namely, 25 years ago Oded Schramm introduced a random version of Loewner’s evolution based on perhaps the most ubiquitous random object: standard Brownian motion.
The random growth process thus obtained — now termed Schramm-Loewner evolution — has been a key player in numerous recent breakthrough results (e.g., computation of the fractal dimension of planar Brownian frontier, proof of conformal invariance of critical planar lattice systems, models for random surfaces and quantum gravity via conformal welding). It has also turned out to share very deep and occasionally surprising connections to other areas of mathematics (e.g., Teichmueller theory, real algebraic geometry, and conformal field theory). I will give a glimpse to this classical topic from the perspective of the 21st century.

## Talks

### Sunil Chhita (Durham University)

**Title: the two-periodic Aztec diamond**

**Abstract:** The two-periodic Aztec diamond is a random tiling model which features three macroscopic regions known as frozen, rough and smooth, which are each characterized by their decay of correlations. In this talk, we will introduce the model as well as describe the behavior at the rough-smooth boundary. This talk is based on joint work with Duncan Dauvergne and Thomas Finn.

### Alessandra Cipriani (University College London)

**Title: Fermionic structure in the Abelian sandpile and the uniform spanning tree**

**Abstract:** In this talk we consider a stochastic system of sand grains moving on a finite graph: the Abelian sandpile, a prototype of self-organized lattice model. We focus on the function that indicates whether a single grain of sand is present at a site, and explore its connections with the discrete Gaussian free field, the uniform spanning tree, and the fermionic Gaussian free field. Based on joint works with L. Chiarini (Durham), R. S. Hazra (Leiden), A. Rapoport and W. Ruszel (Utrecht).

### Lisa Hartung (Johannes Gutenberg University Mainz)

**Title: Branching random walk subject to a hard wall constraint**

**Abstract:** We consider a Gaussian branching random walk under the constraint that the values at all leaves at generation n are non-negative. We obtain a remarkably precise description of the conditional law and the conditional field. The conditioning leads to an upward shift of the whole field. We obtain sharp estimates on this upward shift (up to $o(1)$ terms). We show that the properly rescaled maximum converges to a Gumbel distribution (without a random shift!), and the rescaled minimum is exponentially distributed. We use tools from DGFFs on general graphs and estimates on random walks that are weakly attracted to zero either through a pinning potential or a drift. The talk is based on joint work with M. Fels (Technion) and O. Louidor (Technion).

### Levi Haunschmid-Sibitz (TU Wien)

**Title: Near-critical dimers and massive SLE**

**Abstract:** The uniform dimer model is a classical model from statistical mechanics and one of the few models where conformal invariance has been established. We consider an off-critical weighted version of this model and connect it via Temperley’s bijection and Wilson’s algorithm to a loop-erased random walk. The scaling limit of this walk is (a generalization of) massive SLE$_2$ as constructed by Markarov and Smirnov and might be of independent interest.

In the talk after sketching the connection between the dimer model and the loop-erased random walk, I will focus on this walk and its scaling limit. First I will present some exact Girsanov identities that help connect a random walk with mass with a random walk with drift, and then I will show some of the techniques and difficulties that go into defining the continuum limit.

### Antoine Jego (École Polytechnique Fédérale de Lausanne)

**Title: Uniqueness of Malliavin—Kontsevich—Suhov measures**

**Abstract:** About 20 years ago, Kontsevich & Suhov conjectured the existence and uniqueness of a family of measures on the set of Jordan curves, characterised by conformal invariance and a restriction-type property. This conjecture was motivated by (seemingly unrelated) works of Schramm, Lawler & Werner on Schramm—Loewner evolutions (SLE), and Malliavin, Airault & Thalmaier on “unitarising measures”. The existence of this family was settled by works of Werner—Kemppainen and Zhan, using a loop version of SLE. The uniqueness was recently obtained in a joint work with Baverez.
I will start by reviewing the different notions involved before giving some ideas of our proof of uniqueness: in a nutshell, we construct a family of “orthogonal polynomials” which completely characterises the measure. I will discuss the broader context in which our construction fits, namely the conformal field theory associated with SLE.

### Gaultier Lambert (KTH Royal Institute of Technology)

**Title: Multiplicative chaos for the characteristic polynomial of CβE**

**Abstract:** I will review the coupling for the circular $\beta$-ensembles (C$\beta$E) introduced by Killip and al. and use this framework to describe the asymptotics of powers of the characteristic polynomial and eigenvalue counting function in terms of multiplicative chaos. I will also explain the relationship with previous results on the spectral measure of the corresponding CMV operator. This is joint work with Joseph Najnudel (University of Bristol).

### Léonie Papon (Durham University)

**Title: A level line of the massive Gaussian free field**

**Abstract:**
I will present a coupling between a massive planar Gaussian free field (GFF) and a random curve in which the curve can be interpreted as the level of the field. This coupling is constructed by reweighting the law of the standard GFF-SLE$_4$ coupling. I will then show that in this coupling, the marginal law of the curve is that of a massive version of SLE$_4$, called massive SLE$_4$. This law on curves was orignally introduced by Makarov and Smirnov to describe the scaling limit of a massive version of the harmonic explorer.

### Avelio Sepúlveda (Universidad de Chile)

**Title: Dynamics on the thick points of the Gaussian free field.**

**Abstract:** In this joint work with Felipe Espinosa, I will present two natural dynamics that have the planar Gaussian Free Field (GFF) as an invariant measure, focusing on the behavior of thick points within these dynamics.

The first dynamic is the Ornstein-Uhlenbeck GFF. We demonstrate that the function mapping a point x to its thickness is continuous, and we describe all possible thickness functions that can emerge in this dynamic.

The second dynamic is the solution to the additive stochastic heat equation, where the “thickness function” is discontinuous, allowing for the existence of points that are thicker than $2$, with a maximum thickness of up $2\sqrt{2}$. We investigate the exceptional times in this dynamic—specifically, the times at which points thicker than $2$ appear. For a given thickness $\gamma$, we identify infinitely many phase transitions, corresponding to the existence of exceptional times with more than $N$ points that are $\gamma$-thick. Surprisingly, these phase transitions occur at specific values, including $\gamma^2=8,6,16/3,\dots$.

### Oliver Tough (Durham University)

**Title: The critical fixed point of branching Brownian motion viewed from its tip is unique**

**Abstract:** We prove that the critical fixed point of branching Brownian motion viewed from its rightmost particle is unique, proving a conjecture of Chen, Garban and Shekhar in the critical case.

Moreover we prove that this critical fixed point is the attractor of branching Brownian motion viewed from its rightmost tip for any initial configuration such that the asymptotic velocity is at most the critical velocity $\sqrt{2}$ in the following sense: for any such initial distribution there exists a set of times $\bf{T}$ with asymptotic density $1$ such that we have convergence in distribution to the critical fixed points as $t\rightarrow \infty$ with $t\in \bf{T}$.

The $N$-branching Brownian motion with selection ($N$-BBM) belongs to a class of branching-selection particle systems introduced by Brunet and Derrida in the physics literature. We prove that the $N$-BBM viewed in its stationary distribution from its rightmost particle converges as $N\rightarrow \infty$ to this critical fixed point of branching Brownian motion viewed from its tip. We explain why we expect the same to be true in general for branching-selection particle systems in the pulled (but not the pushed) regimes.

This is work in progress.

### Wendelin Werner (University of Cambridge)

**Title: Loop-soup rewiring dynamics and** $\Phi^4$ **fields**

**Abstract:** It is now well-known how to construct the GFF from Brownian
loop-soups that are collections of “independent” Brownian loops. I will try to
explain how to tweak this construction to obtain the $\Phi^4$ fields via
Brownian loops interacting via their intersection local times. Ideas coming
from the loop-soup rewiring dynamics turn out to be fruitful.

## Short talks

### Luis Brummet (Aalto University)

**Title: Complex Driven Loewner Equations**

**Abstract:**
The Loewner differential equation, originally introduced by Charles Loewner in 1923, is a differential equation that is extensively studied in complex analysis as well as in probability theory. One can see it as the deterministic foundation of so called ”Schramm-Loewner evolutions (SLE)”, which have countable applications in statistical physics, most notably in the study of critical phenomena of two dimensional conformally invariant models. Roughly speaking, both the deterministic as well as the probabilistic side the Loewner equations allow to encode geometries of two dimensional sets by using a so called driving function. Historically only real valued driving functions where considered, however in recent years Tran [Tra17] started the study of complex valued driving functions. Following Tran, Lind [LU22] built two classification results about the geometry arising from a one parameter family of driving functions. We give an introduction to complex driven Loewner chains from the deterministic side by considering the known theory and then present a classification result of the complex linear driver and its relationship to holomorphic motion and the results done by Tran and Lind.

### Harry Giles (Warwick University)

**Title: Self-repelling Brownian Polymer in the critical dimension**

**Abstract:** The Self-repelling Brownian Polymer is a type of weakly self-avoiding motion in $mathbb{R}^d$ that was introduced separately by physicists (Amit, Parisi, Peliti, ’83) and probabilists (Norris, Rogers, Williams, ’87). In dimension $d = 3$ and higher, there is an invariance principle: look at large diffusive scales, and you will see a Brownian motion. However, in the critical dimension, $d = 2$, the process is logarithmically super-diffusive, making it harder to study. Nonetheless, we can show that one still sees Brownian motion at large scales, provided that the rescaling is done in the “weak coupling” sense, whereby as we zoom out, we also tune down the strength of self-interaction. Strangely enough, while this is fundamentally a non-Markovian problem, our techniques are Markovian in nature.

### Akshay Hegde (University of Oxford)

**Title: Volume of level sets of smooth Gaussian fields**

**Abstract:** Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F = f_1 −f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis. This is joint work with Dmitry Beliaev.

### Remy Mahfouf (University of Geneva)

**Title: Weak IID disorder preserve criticality of the Ising model, even beyond the critical window**

**Abstract:** We prove Russo-Seymour-Welsh type crossing estimates for the FK-Ising model on general s-embeddings whose origami map has an asymptotic Lipschitz constant strictly smaller than 1, provided a mild non-degeneracy assumption is satisfied. This result extends the original work of Chelkak and provides a general framework to prove that connection probabilities between boundaries of boxes remain bounded away from 0 and 1. This can be applied to derive criticality of the near critical Ising model on the square lattice with IID random coupling constant around the critical point, with a typical deviation that would make the model off critical if it was deterministic instead of random.

### Anne Schreuder (University of Cambridge)

**Title: Lévy-driven Schramm-Loewner Evolutions**

**Abstract:** This talk is about the behaviour of Loewner evolutions driven by a Lévy process. Schramm’s celebrated version (Schramm-Loewner evolution), driven by standard Brownian motion, has been a great success for describing critical interfaces in statistical physics. Loewner evolutions with other random drivers have been proposed, for instance, as candidates for finding extremal multifractal spectra, and some tree-like growth processes in statistical physics. Questions on how the Loewner trace behaves, e.g., whether it is generated by a (discontinuous) curve, whether it is locally connected, tree-like, or forest-like, have been partially answered in the symmetric alpha-stable case. We consider the case of general Levy drivers. Joint work with Eveliina Peltola (Bonn and Helsinki).

### Yizheng Yuan (University of Cambridge)

**Title: The chemical distance metric for non-simple CLE**

**Abstract:** I will introduce the continuum analogue of the chemical distance metric in lattice models such as percolation. The chemical distance metric is the graph distance induced by the percolation clusters. It is known that for critical percolation, the lengths have non-trivial scaling behaviour, however it is very difficult to find the exact scaling exponent. (This is one of the questions from Schramm’s ICM 2006 article that remains unsolved.)

In a joint work with Valeria Ambrosio and Jason Miller, we construct a chemical distance metric on the CLE gasket for each $\kappa \in ]4,8[$. We show that it is unique metric that is geodesic, Markovian, and conformally covariant. The proof is reminiscent to the construction of the LQG metric, but our objects behave very differently, and hence our techniques also differ significantly from those used in LQG. For $\kappa=6$, we conjecture that our random metric space is the scaling limit of critical percolation.

# Practical information

## Getting to Durham

Durham is well connected by trains, for live information please see Trainline webpage.

Train connections to other cities with airports:

- London: about 3 hours by train (from King’s Cross train station)
- Edinburgh: about 2 hours by train
- Newcastle: about 20 minutes by train

For more information, see Durham University webpage on the subject.

## Getting to the Department

The Department of Mathematical Sciences is about a 37-minute walk from Durham train station. There are also taxis available at the train station. For more information see the map below.

## Where to eat

Durham has many nice restaurants including

- Whitechurch free house: Pub close to the department
- Zen: Thai/pan-asian restaurant
- Nadon: Thai restaurant
- Akarsu: Turkish restaurant
- Lebaneat: Lebanese restaurant
- Fat Hippo Durham: Burgers and chips

# About the Durham Symposium

## Event’s history

Mathematics symposia have been held in Durham every year since 1974, and have since become an established and recognised series of international research meetings, with over 100 symposia to date. They provide an excellent opportunity to explore an area of research in depth, to learn of new developments, and to instigate links between different branches. The format is designed to allow substantial time for interaction and research. The week-long meetings are held in July or August, with about 50 participants, roughly half of whom will come from the UK. Lectures and seminars take place in the Department of Mathematical Sciences, Durham University.

## Organizing Committee

## Funding

This event is supported by Durham University, the EPSRC, and the London Mathematical Society.