We study the tail probability of the mass of Gaussian multiplicative chaos and establish a formula for the leading order asymptotics under very mild assumptions, resolving a recent conjecture of Rhodes and Vargas. The leading order coefficient can be described by the product of two constants, one capturing the dependence on the test set and any non-stationarity and the other one encoding the universal properties of multiplicative chaos. This may be seen as a first step in understanding the full distributional properties of Gaussian multiplicative chaos.

This is the first talk in a five part series of talks on deep learning from a theoretical point of view, held jointly between the probability theory and machine learning groups of the Department of Mathematics and Computer Science. The four invited speakers that follow after this talk are young researchers who are contributing in different ways to what will hopefully eventually be a comprehensive theory of deep neural networks.

In this first talk I will introduce the main theoretical questions about deep neural networks:
1. Representation - what can deep neural networks represent?
2. Optimization - why and under what circumstances can we successfully train neural networks?
3. Generalization - why do deep neural networks often generalize well, despite huge capacity?

As a preface I will review the basic models and algorithms (Neural Networks, (stochastic) gradient descent, ...) and some important concepts from machine learning (capacity, overfitting/underfitting, generalization, ...).

An excursion around the ideas for why the stochastic gradient descent algorithm works well on training deep neural networks leads to considerations about the underlying geometry of the related loss function. Recently, we gained a lot of insight into how tuning SGD leads to better or worse generalization properties on a given model and task. Furthermore, we have a reasonably large set of observations that lead to the conclusion that more parameters typically lead to better accuracies as long as the training process is not hampered. In this talk, I will speculatively argue that as long as the model is over-parameterized (OP), all solutions are equivalent up to finite size fluctuations.
We will start by reviewing some of the recent literature on the geometry of the loss function, and how SGD navigates the landscape in the OP regime. Then we will see how to define OP by finding a sharp transition described by the models fitting abilities to its training set. Finally, we will discuss how this critical threshold is connected to the generalization properties of the model, and argue that life beyond this threshold is (more or less) as good as it gets.

We show that the behaviour of a Deep Neural Network (DNN) during gradient descent is described by a new kernel: the Neural Tangent Kernel (NTK). More precisely, as the parameters are trained using gradient descent, the network function (which maps the network inputs to the network outputs) follows a so-called kernel gradient descent w.r.t. the NTK. We prove that as the network layers get wider and wider, the NTK converges to a deterministic limit at initialization, which stays constant during training. This implies in particular that if the NTK is positive definite, the network function converges to a global minimum. The NTK also describes how DNNs generalise outside the training set: for a least squares cost, the network function converges in expectation to the NTK kernel ridgeless regression, explaining how DNNs generalise in the so-called overparametrized regime, which is at the heart of most recent developments in deep learning.

The current successes achieved by neural networks are mostly driven by experimental exploration of various architectures, pipelines, and hyper-parameters, motivated by intuition rather than precise theories. Focusing on the optimization/training aspect, we will see in this talk why pushing theory forward is challenging, but also why it matters and key insights it may lead to. We will review some recent results on the phenomenon of "lazy training", on the role of over-parameterization, and on training neural networks with a single hidden layer.

The complexity of deep neural networks remains an obstacle to the understanding of their great efficiency. Their generalisation ability, a priori counter intuitive, is not yet fully accounted for. Recently an information theoretic approach was proposed to investigate this question.
Relying on the heuristic replica method from statistical physics we present an estimator for entropies and mutual informations in models of deep model networks. Using this new tool, we test numerically the relation between generalisation and information.

TBA

The classical random walk isomorphism theorems relate the local time of a random walk to the square of a Gaussian free field. I will present non-Gaussian versions of these theorems, relating hyperbolic and hemispherical sigma models (and their supersymmetric versions) to non-Markovian random walks interacting through their local time. Applications include a short proof of the Sabot-Tarres limiting formula for the vertex-reinforced jump process (VRJP) and a Mermin-Wagner theorem for hyperbolic sigma models and the VRJP. This is joint work with Tyler Helmuth and Andrew Swan.

In this talk we will study the asymptotic behavior of a random walk that evolves on top of a simple symmetric exclusion process. This nice example of a random walk on a dynamical random environment presents its own challenges due to the slow mixing properties of the underlying medium. We will discuss a law of large numbers that has been proved recently for this random walk. Interestingly, we can only prove this law of large numbers for all but two exceptional densities of the exclusion process. The main technique that we have employed is a multi-scale renormalization that has been derived from works in percolation theory.

In this talk, we consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the partition function. It was proven by Comets and Liu that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole L^2-region, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. This is joint work with Clément Cosco.

It was proven by Rider and Virag that the logarithm of the characteristic polynomial of the Ginibre ensemble converges to a logarithmically correlated random field. In this talk we will see how this connection can be established on the level if powers of the characteristic polynomial by proving convergence to Gaussian multiplicative chaos. We consider the range of powers in the \(L^2\) phase.
(Joint work in progress with Paul Bourgade and Guillaume Dubach).

We consider two fundamental percolation models with long-range correlations: The Gaussian free field and (the vacant set) of Random Interlacements. Both models have been the subject of intensive research during the last years and decades, on \(\mathbb Z^d\) as well as on some more general graphs. We investigate some structural percolative properties around their critical parameters, in particular the ubiquity of the infinite components of complementary phases.
This talk is based on joint works with A. Prévost (Köln) and P.-F. Rodriguez (Bures-sur-Yvette).

Finding the (space-height) distribution of the (local) extrema of high-dimensional strongly correlated random fields is a notorious hard problem with many applications. Following Fyodorov and Sommers (2007), we focus on the Gaussian fields with isotropic increments and take the viewpoint of statistical physics. By exploiting various probabilistic symmetries, we rigorously derive the Fyodorov-Sommers formula for the log-partition function in the high-dimensional limit. The formula suggests a rich picture for the distribution of the local extrema akin to the celebrated spherical Sherrington-Kirkpatrick model with mixed p-spin interactions.

For Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type.

Consider the hypercube as a graph with vertex set \({0,1}^N\) and edges between two vertices if they are only one coordinate flip apart. Choosing independent standard exponentially distributed lengths for all edges and asking how long the shortest directed paths from \((0,..,0)\) to \((1,..,1)\) is defines oriented first passage percolation on the hypercube. We will discuss the conceptual steps needed to answer this question to the precision of extremal process following the two paper series "Oriented first passage percolation in the mean field limit" by Nicola Kistler, Adrien Schertzer and Marius A. Schmidt: arXiv:1804.03117 [math.PR] and arXiv:1808.04598 [math.PR].

We consider the adjacency matrix of the Erdos-Renyi graph \(G(N,p)\) in the supercritical regime \(pN > C \log N\) for some universal constant C. We show that the eigenvalue density is with high probability well approximated by the semicircle law on all spectral scales larger than the typical eigenvalue spacing. We also show that all eigenvectors are completely delocalized with high probability. Both results are optimal in the sense that they are known to be false for \(pN < \log N\). A key ingredient of the proof is a new family of large deviation estimates for multilinear forms of sparse vectors. Joint work with Yukun He and Matteo Marcozzi.

The goal of this talk is to explain how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations. These are known as “rigidity estimates” in the literature and they play a crucial role in the proof of universality. I will review some of the current results on eigenvalues’ fluctuations and present a new approach which relies on the theory of Gaussian Multiplicative Chaos and leads to optimal rigidity estimates for the Gaussian Unitary Ensemble. I will also mention how it is also deduce a central limit theorem from our proof. This is joint work with Tom Claeys, Benjamin Fahs and Christian Webb.

Uniform integer-valued Lipschitz functions on a finite domain of the triangular lattice are shown to have variations of logarithmic order in the radius of the domain. The level lines of such functions form a loop \(O(2)\) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at 0; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model. Based on joint work with Alexander Glazman.