The concept of self-organized criticality was coined in to describe some physical systems which present a critical-like behaviour, without the need to finely tune a parameter (like for example the temperature) to a particular value. This idea has been put forward in various real-life settings (sand piles, forest fires, avalanches, neural networks, earthquakes...), with more or less controversial outcomes. On the theoretical side, the rigorous construction and analysis of a mathematical model showing self-organized criticality is a difficult problem, and even models whose definition is very simple, like the sandpile model, are not well understood.

In this talk, I will present several toy models of self-organized criticality built upon percolation and the Ising model by introducing a feedback mechanism from the configuration onto the control parameter. The study of these models, which is a nice occasion to play around near-critical phenomena, will lead us to compare with other related models and to discuss about the ingredients necessary to obtain a self-critical behaviour.

Bipartite spin glasses -- a variant of usual spin glasses, in which spins are grouped into two species -- are a classical testing ground for new approaches in spin glasses. The annealed complexity of bipartite spherical models was initially considered by Auffinger and Chen, who gave upper and lower bounds. We give an exact variational formula for this complexity, both for pure spin glasses and for mixtures. We also find connections between this model and the usual spherical spin glasses, which were studied by Auffinger, Ben Arous, and Černý.

Bipartite spin glass models have been gaining popularity in the study of glassy systems with distinct interacting species. Recently, the annealed complexity of the pure spherical bipartite model was obtained by B. McKenna. In this talk, I will explain how to show that the low-lying complexity actually concentrates around this value, and how from this one can obtain a formula for the ground-state energy.

In this talk, I will introduce how we can use the Kac-Rice method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents the first application of Kac-Rice to obtain exact complexity asymptotics for non-Gaussian random functions. We obtain a rigorous explicit variational formula for the average number of critical points. This result is then extended using the non-rigorous Kac-Rice replicated method from theoretical physics. In this way we derive a second variational formula for the number of critical points for typical instances up to exponential accuracy. Finally, I will discuss some of the challenges ahead, in the numerical evaluation and extension of these formulas. This talk is based on the results of the paper https://arxiv.org/abs/1912.02143.

TBA.

Based https://arxiv.org/abs/2107.06123 and https://arxiv.org/abs/2111.15577.

This talk will focus on results from a recent paper by Baik, Collins-Woodfin, Le Doussal and Wu. In the paper, we analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick (SSK) spin glass model with an external field. Our goal is to understand the transition between this model and the one without an external field. My talk will provide a brief introduction to the SSK model, focusing on the distribution of spins (i.e. the geometry of the Gibbs measure) and how the spin distribution changes in the presence of an external field. I will then discuss the transitional case in which the strength of the external field goes to zero as the dimension of the spin variable grows. I will present results for overlaps with the external field, with the ground state and with a replica, focusing on what each of these overlaps tells us about the distribution of spins. Finally, I will provide a brief overview of our methods, including a contour integral representation of the partition function as well as random matrix techniques.

The Fisher-KPP equation is a well studied reaction-diffusion equation motivated by biology as well as statistical physics. The solutions exhibit traveling "waves" that propagate through space with a finite speed. Propagating waves also exist for solutions to the stochastic Fisher-KPP equation (where the original FKPP equation is perturbed by a white noise term). We determine the asymptotic behavior of the wave-speed when the coefficient determining noise-strength approaches zero.

Several studies on heritability in twins aim at understanding the different contribution of environmental and genetic factors to specific traits. Considering the National Merit Twin Study, our purpose is to correctly analyse the influence of the socioeconomic status on the relationship between twins’ cognitive abilities. Our methodology is based on conditional copulas, which allow us to model the effect of a covariate driving the strength of dependence between the main variables. We propose a flexible Bayesian nonparametric approach for the estimation of conditional copulas, which can model any conditional copula density. Our methodology extends previous work by introducing dependence from a covariate in an infinite mixture model. Our results suggest that environmental factors are more influential in families with lower socio-economic position.

In this talk, I will review the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass and present a dynamical derivation, valid at sufficiently high temperature. In our derivation, the TAP equations follow as a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions which implies an analogue of the TAP equations for the two point functions. The talk is based on joint work with A. Adhikari, P. von Soosten and H.T. Yau.

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms. We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish the contiguity conjecture between the planted and unplanted models in the satisfiable regime and other properties of the structure of the solution space. Our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

Over the last 20 years the field of Epidemiology has embraced the exploitation of random genetic inheritance to help uncover causal mechanisms of disease using the technique of Mendelian randomization (MR). Genetic variants can also play an important role in helping to explain treatment effect heterogeneity, through the science of pharmacogenetics. A canonical example is Clopidogrel: the primary drug for stroke prevention in the UK and many other countries. It requires CYP2C19 enzyme activation in order to be properly metabolised and thus work to its fullest extent. We review the methodological underpinnings of the general pharmaco-genetic approach, which utilises only individuals who are treated and relies on fairly strong baseline assumptions to estimate what we refer to as the `genetically mediated treatment effect' (GMTE). When these assumptions are seriously violated, we show that a robust estimate of the GMTE that incorporates information on the population of untreated individuals can instead be used. In cases of partial violation, we clarify when Mendelian randomization and a modified confounder adjustment method can also yield consistent estimates for the GMTE. A full decision framework is then described to decide when a particular estimation strategy is most appropriate and how estimates can be combined to improve efficiency. We illustrate these approaches by re-analysing UK Biobank-CPRD linked data relating to CYP2C19 genetic variants, Clopidogrel use and stroke risk, and data relating to ApoE genetic variants, statin use and Coronary Artery Disease. We then discuss how the framework can be extended for use in other epidemiological contexts.

This work focuses on the extension of the Parisi full replica symmetry breaking solution to the Ising spin glass on a random regular graph. We propose a new martingale approach that overcomes the limits of the Parisi-Mézard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. We obtain a variational free energy functional, defined by the sum of two variational functionals (auxiliary variational functionals), that are an extension of the Parisi functional of the Sherrington-Kirkpatrick model. We study the properties of the two variational functionals in detail, providing a representation through the solution of a proper backward stochastic differential equation, that generalize the Parisi partial differential equation. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and the free energy.

The Gaussian Free Field, usually studied on transient graphs, can also be defined on finite graphs (its covariance is given by a compensated Green function). This talk is about the GFF on typical regular graphs whose number of vertices goes to infinity. We will focus on the percolation of the GFF above a fixed level \(h\). If \(h\) is below a certain critical threshold \(h_*\), we establish the emergence w.h.p. of a unique giant component containing a positive proportion of the vertices, surrounded by islets of logarithmic size, while only the latter survive if \(h\ge h*\). This big continent shares many similarities with the giant component of the famous Erdős-Rényi random graph.

In our talk, we will consider random point processes formed by the clusters in one- dimensional coalescing particle systems and connected to the real Ginibre ensemble. Basing on recent work of R. Tribe and O. Zaboronski, we will first discuss their Pfaffian structure in the Brownian case, and then, by proposing a new algebraic proof, prove their conjecture that the obtained results remain valid in the general case.
Furthermore, we will speak about the concentration of measure in diffeomorphic approximations of Brownian coalescing particle systems. In particular, the high-level exceedances of the stationary processes formed by the corresponding densities will be studied. More precisely, using the Rice formula, we will compute the level-crossing intensity of these processes and establish its asymptotic behaviour as the height of the level tends to infinity.

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the relaxation time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

Questions on the treatment effect in sub-populations defined by post-randomization events, are not uncommon in drug development. In the causal inference literature these are typically referred to as principal stratum estimands. As these populations are defined based on events that may be influenced by the received treatment, randomization alone can no longer be relied upon for assessment of the treatment effect. Additional assumptions are required to identify the estimand. As an example consider a multiple sclerosis trial, where interest may be in the treatment effect on confirmed disability progression in the population that does not relapse during the trial. When the treatment prevents relapses, it is difficult to perform this assessment, because the non-relapser populations on treatment and control group may not be comparable. As assumptions play an important role for identification of principal stratum estimands, sensitivity analyses should play a crucial role too. I will review different approaches in this area and illustrate them in the context of a simulated example data set and a recently proposed Bayesian analysis strategy.

The realm of statistical mechanics has been enlarged to describe systems, such as glass forming materials, where structural disorder plays the predominant role. Interestingly the spectrum of applications of this new physics goes much beyond the scope of condensed matter and extends to the currently booming field of data science. In this talk I will focus on the challenge of signal-reconstruction from noisy collections of data, omnipresent in machine learning applications and in classical inference problems. By leveraging tools and ideas from glass physics, I will give some examples on how we can describe, predict, and enhance the performances of algorithms introduced to tackle these reconstruction problems

TBA

In this talk we focus on a class of disorder systems characterized by the spherical pure p-spin Hamiltonian in the presence of a nonlinear signal given by a deterministic polynomial of degree k. (We will call them the (p,k) spiked tensor model). It is known recently that these systems exhibit phase transitions in terms of the topology of their energy landscape. We characterize the mean number of deep minima near the bottom of the landscape through the Kac-Rice formula. When the strength of the signal is beyond a conjectured trivialization threshold, the mean number of deep minima is asymptotically finite (but is surprisingly not equal to 1) and we derive an explicit formula for the limiting ground states and the limiting ground state energy by combining techniques in spin glass theory and random matrices. This talk is based on joint work with Antonio Auffinger and Gerard Ben Arous.

The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations) and elastic self-interactions (preferring ordered configurations). The elastic manifold model displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method. We study the topology of the energy landscape of this model in the Mézard-Parisi setting, computing the (annealed) topological complexity both of total critical points and of local minima. Our results confirm recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The argument relies on asymptotics of large random determinants beyond invariant random matrices, and the analysis of the associated Matrix Dyson Equations. This is based on joint work with Gérard Ben Arous and Ben McKenna.

We will consider two problems, which are related to some projects with F.Hoffman-Roche. The first one concerns the application of Stochastic Differential Equations (Ito’s Integrals) in pricing problems. The second is about mixtures of extreme value distributions studied with techniques from algebraic statistics.

I will discuss the complexity of minima and saddles in two classic examples of random landscapes: the spherical p-spin glass model and the model of a single particle in a random confining potential. In these models, the number of critical points diverge exponentially as the dimension of the ambient space grows. By combining Kac-Rice’s formula and random matrix tools, we are able to estimate this exponential growth. I will also explain some consequences of these results to statistical physics and optimization theory.

In this talk, I will illustrate how critical decisions in drug development are typically based on a tiny fraction of the collected data only. As an example, in early development oncology clinical trials, the decision whether to move a molecule to Phase 3 is typically based on response proportions and duration of response in those that respond, while in Phase 3 the primary endpoint will be long-term endpoints such as progression-free (PFS) or overall survival (OS). Effects on response-based short-term endpoints seldom translate in effects on these relevant endpoints. We propose to make decisions not based on intermediate endpoints, but on a prediction of the OS hazard ratio (HR) between data of the new molecule collected in the early phase trial and historical data of the control treatment. This HR prediction is using a multistate model based on the various disease states a patient may go through until death. This yields a gating strategy with improved operating characteristics compared to traditional decision rules in the context of early phase clinical trials. If time permits I will further discuss how the joint distribution of PFS and OS as a function of transition probabilities in a multistate model can be derived. No assumptions on copulae or latent event times are needed and the model is allowed to be non-Markov. From the joint distribution, statistics of interest can then readily be computed. As an example, we provide closed formulas and statistical inference for Pearson’s correlation coefficient between PFS and OS. Our proposal complements existing approaches by providing methods of statistical inference while at the same time working within a much more parsimonious modelling framework. The main conclusion of this talk is that multistate models are a useful and underutilized tool in the analysis of clinical trial data.

Many studies aim to assess treatment effects on outcomes in individuals characterized by status on a particular post-treatment variable. For example, we may be interested in the effect of cancer therapies on quality of life, and quality of life is only well-defined in the those individuals who are alive. Similarly, we may be interested in the effect of vaccines on post-infections outcomes, which are only of interest in those individuals who become infected. In these settings, a naive contrast of outcomes conditional on the post-treatment variable does not have a causal interpretation, even in a randomized experiment. Therefore the effect in the principal stratum of those who would have the same value of the post-treatment variable regardless of treatment, such as the survivor average causal effect, is often advocated for causal inference. Whereas this principal stratum effect is a well defined causal contrast, it cannot be identified without strong untestable assumptions, and its practical relevance is ambiguous because it is restricted to an unknown subpopulation of unknown size. Here we formulate alternative estimands, which allow us to define the conditional separable effects. We describe the causal interpretation of the conditional separable effects, e.g. in settings with truncation by death, and introduce three different estimators. As an illustration, we use data from a randomized clinical trial to estimate a conditional separable effect of chemotherapies on quality of life in patients with prostate cancer.

Given a probability measure measure on the discrete hypercube, we describe a method to derive sufficient conditions for the measure to admit a "pure state decomposition", roughly speaking, to be expressed as a mixture of a small number of "localized" measures. Such a decomposition theorem has applications to mean-field approximation of Gibbs measures and to random graphs. Our method is based on the construction of a stochastic process which samples from the measure.

A generalization of the Sherrington-Kirkpatrick (SK) model will be introduced, where spin variables are arranged over K layers and only consecutive layers interact. From the Machine Learning point of view this model can be thought as the random version of a deep Boltzmann machine. A lower bound for the quenched pressure will be shown in terms of K SK-models. Then suitable conditions on the layers temperatures and sizes will be provided in order to guarantee that the quenched pressure coincides with the annealed one or satisfies a replica symmetric bound.

The celebrated Thouless-Anderson-Palmer approach suggests a way to relate the free energy of a mean-field spin glass model to the solutions of certain self-consistency equations for the local magnetizations. I will describe a new geometric approach to define free energy landscapes for general spherical mixed p-spin models and derive from them a generalized TAP representation for the free energy. Time permitting, I will explain how these landscapes are related to the pure states decomposition, ultrametricity property, and optimization of full-RSB models.

I will give a broad overview of techniques in resurgent analysis and transseries, as they apply in string theory and matrix models (and with some focus on Painlevé type equations).

I will discuss a generalization of the classic condition of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the L^2 metric structure of the Gaussian random variables. As an example of application I will deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail. I will underline the dependence of the density of quenched free energy just on the metric structure and discuss the models from a metric viewpoint. Joint work with Roberto Boccagna.

We consider a spin system containing pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by Baik and Lee and Baik et al. which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to Baik and Lee in the “paramagnetic regime” when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by Banerjee . The proof of the approximation by the linear spectral statistics part is close to Banerjee and Ma .

In this talk, I will illustrate how critical decisions in drug development are typically based on a tiny fraction of the collected data only. As an example, in early development oncology clinical trials, the decision whether to move a molecule to Phase 3 is typically based on response proportions and duration of response in those that respond, while in Phase 3 the primary endpoint will be long-term endpoints such as progression-free (PFS) or overall survival (OS). Effects on response-based short-term endpoints seldom translate in effects on these relevant endpoints. We propose to make decisions not based on intermediate endpoints, but on a prediction of the OS hazard ratio (HR) between data of the new molecule collected in the early phase trial and historical data of the control treatment. This HR prediction is using a multistate model based on the various disease states a patient may go through until death. This yields a gating strategy with improved operating characteristics compared to traditional decision rules in the context of early phase clinical trials. If time permits I will further discuss how the joint distribution of PFS and OS as a function of transition probabilities in a multistate model can be derived. No assumptions on copulae or latent event times are needed and the model is allowed to be non-Markov. From the joint distribution, statistics of interest can then readily be computed. As an example, we provide closed formulas and statistical inference for Pearson’s correlation coefficient between PFS and OS. Our proposal complements existing approaches by providing methods of statistical inference while at the same time working within a much more parsimonious modelling framework. The main conclusion of this talk is that multistate models are a useful and underutilized tool in the analysis of clinical trial data.

This is joint work with Ulrich Beyer, Jan Beyersmann, Matthias Meller, David Dejardin, and Uli Burger.

Spherical spin glass models can be interpreted as the optimization of a random polynomial on a sphere in high dimensions. Computing the free energy will give a formula for the typical maximum value of this random process as the dimension goes to infinity. These models have appeared in areas such as physics, statistics, and computer science. In this talk, we will present a free energy formula for coupled systems of spherical spin glasses. We will describe several of the main mathematical ideas in the derivation of this formula such as ultrametricity and synchronization.

We analyse the metastable behaviour of the dilute Curie–Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are replaced by i.i.d. random coefficients, e.g. Bernoulli random variables with fixed parameter p. This model can be also viewed as an Ising model on the Erdos–Renyi random graph with edge probability p. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature \beta. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where the system size goes to infinity, the inverse temperature is larger than 1 and the magnetic field is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie–Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities.

The clique-separator factorization of the joint distribution of random variables in decomposable graphical models allows for conjugate choices for the prior distributions of both parameters and graphical structure. I will describe a Markov chain Monte Carlo approach for graphical model estimation when the parameter prior has the Hyper Markov property and the graph prior is Structural Markov. I will focus on computational considerations and consider extensions to the Weak Structural Markov case.

Characterizing the geometrical properties of non-convex, high-dimensional landscapes is a longstanding problem in glassy systems, which is now gaining increasing centrality in the fields of inference, machine learning, ecology and biology. Among these properties, an important role is played by the statistics of the stationary points (local minima, maxima and saddles), which is a crucial ingredient to understand how the landscape is explored and optimized with local dynamics. In this talk I will summarize some recent progress done in the analysis of a prototypical random landscape, the energy functional of the so-called spherical p-spin model or random Gaussian monomial. I will discuss how to extract information on the connectivity of sub-optimal minima in configuration space and on the distribution of the energy barriers separating them, using techniques of random matrix theory. I will then comment on the dynamical implications of these results on the dynamics, focusing in particular on the activated regime.

In this talk I will present a variational formula for the thermodynamic limit of the free energy of the Sherrington-Kirkpatrick (SK) spin-glass model with transverse field, a quantum generalization of the classical SK model. Through its path integral representation, the quantum SK model can be translated to a classical vector-spin model with the spins taking values in the space of $ \{ -1, 1 \} $-valued cadlag paths. I will explain how to approximate the model by a sequence of classical finite-dimensional vector-spin models which enables us to use results of Panchenko to determine the free energy in the thermodynamic limit. The talk is based on joint work with Arka Adhikari.

Approximate Message Passing (AMP) algorithms are non-linear power iteration methods originally arising from the context of compressed sensing. In this talk, I will introduce a Lipschitizian functional iteration, as a generalization of the AMP algorithms, and discuss its universality in disorder. In addition, I will explain how our results imply universality in a number of AMPs popularly adapted in Bayesian inferences and optimizations in spin glasses.

In this talk we will discuss some recent large deviation asymptotics concerning the local behavior of random interlacements on \(\mathbb Z^d\), \(d\ge 3\). In particular, we will describe the link with previous results concerning macroscopic holes left inside a large box, by the the adequately thickened connected component of the boundary of the box in the vacant sets of random interlacements.

We will discuss recent progress regarding the geometry of the Gaussian free field in three and more dimensions. Our results, based on joint work with H. Duminil-Copin, S. Goswami and F. Severo, deal with the percolation problem associated to level-sets of the Gaussian free field, as first investigated by J. Lebowitz and H. Saleur in 1986. We will examine the equality of several natural critical parameters related to this model.

Among percolation models with long-range correlations, the Gaussian free field on discrete graphs has received a lot of attention over the last few years, but not so much on its continuous equivalent, the cable system. I will present a brief history of the question, and explain why, for this model, the critical parameter is surprisingly equal to 0, and is thus explicitly known, on a large class of graphs. There are different proofs of this result, through uniqueness of the infinite cluster, Russo formula, exploration martingale or random interlacements, but also examples where the critical parameter is not equal to 0. Finally, I will give the law for the capacity of the clusters of the level sets of the Gaussian free field on the cable system.
Joint work with Alexander Drewitz and Pierre-François Rodriguez.

I will give an introduction to random tensors and their applications. In particular I will describe an universality result for invariant probability measures for tensors: under generic scaling assumptions, for large tensor sizes any invariant tensor measure approaches a Gaussian. I will then discuss the implications of this result, as well as ways to avoid it.

Mean-field methods fail to reconstruct the parameters of the model when the dataset is clusterized. This situation is found at low temperatures because of the emergence of multiple thermodynamic states. The paradigmatic Hopfield model is considered in a teacher-student scenario as a problem of unsupervised learning with Restricted Boltzmann Machines (RBM). For different choices of the priors on units and weights, the replica symmetric phase diagram of random RBM’s is analyzed and in particular the paramagnetic phase boundary is presented as directly related to the optimal size of the training set necessary for a good generalization. The connection between the direct and inverse problem is pointed out by showing that inference can be efficiently performed by suitably adapting both standard learning techniques and standard approaches to the direct problem.

High-dimensional generalized linear models are basic building blocks of current data analysis tools including multilayers neural networks. They arise in signal processing, statistical inference, machine learning, communication theory, and other fields. I will explain how to establish rigorously the intrinsic information-theoretic limitations of inference and learning for a class of randomly generated instances of generalized linear models, thus closing several old conjectures. Examples will be shown where one can delimit regions of parameters for which the optimal error rates are efficiently achievable with currently known algorithms. I will discuss how the proof technique, based on the recently developed adaptive interpolation method, is able to deal with the output nonlinearity and also to some extent with non-separable input distributions.

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.