Probability and Mathematical Physics Seminar

Synopsis:

In the classical finite dimensional setting, a Gaussian multiplicative chaos (GMC) is obtained by tilting an ambient measure by the exponential of a centred Gaussian field indexed by a domain in the Euclidean space. In the two-dimensional setting and when the underlying field is "log-correlated", GMC measures share close connection to the 2D Liouville quantum gravity, which has seen a lot of revived interest in the recent years. 

A natural question is to construct a GMC in the infinite dimensional setting, where techniques based on log-correlated fields in finite dimensions are no longer available. In the present context, we consider a GMC on the classical Wiener space, driven by a (mollified) Gaussian space-time white noise. In $d\geq 3$, in a previous work with A. Shamov and O. Zeitouni, we showed that the total mass of this GMC, which is directly connected to the (smoothened) Kardar-Parisi-Zhang equation in $d\geq 3$, converges for small noise intensity to a well-defined strictly positive random variable, while for larger intensity (i.e. for small temperature) it collapses to zero. We will report on joint work with Yannic Bröker (Münster) where we study the endpoint distribution of a Brownian path under the GMC measure and show that, for low temperature, the endpoint GMC distribution localizes in few spatial islands and produces asymptotically purely atomic states.

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This is a joint work with Jean Barbier, Florent Krzakala, Nicolas Macris and Lenka Zdeborova.

We consider generalized linear models (GLMs) where an unknown $n$-dimensional signal vector is observed through the application of a random matrix and a (non-linear)  componentwise output function.

We study the models in the high-dimensional limit, where the observation consists of $m$ points, and $m/n \to \alpha > 0$ as $n \to \infty$. This situation is ubiquitous in applications ranging from supervised machine learning to signal processing.

We will observe some phase transition phenomena. Depending on the noise level, the distribution of the signal and the non-linear function of the GLM we may encounter various scenarios where it may be possible or not to recover the signal.

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Synopsis:

• 9:30-10:00 am  Registration and Refreshments

• 10:00-11:00 am Rafal Latala (University of Warsaw)
  "Strong and weak moments of random vectors"

• 11:00-11:30 am Refreshments

• 11:30-12:30 am Oren Louidor  (Technion)
  "A scaling limit for the cover time of the binary tree"

• 12:30 - 2:00 pm  Lunch

• 2:00 -  3:00 pm  Junior participant talks

• 3:00 -  3:30 pm  Refeshments

• 3:30 -  4:45 pm  Junior participant talks

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• 9:30 - 10:00 am  Registration and Refreshments
• 10:00 -11:00 am  Patricia Goncalves  (Instituto Superior Técnico)
  "Deriving the regional fractional Laplacian with several boundary conditions."
• 11:00-11:30 am Refreshments
• 11:30-12:30 am   Gabor Lugosi (Fabra University)
  "Noise sensitivity of the top eigenvector of a Wigner matrix."
• 12:30 - 2:00 pm  Lunch
• 2:00 -  3:00 pm  Junior participant talks
• 3:00 -  3:30 pm Refeshments
• 3:30 -  4:45 pm  Junior participant talks
• 5:00 - 7:00 pm  Conference Reception and Dinner

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Synopsis:

We consider a simple idea to decompose log-correlated Gaussian fields into two parts, both of which behave well in a suitable sense. Applications include Onsager type inequalities in all dimensions, analytic dependence and existence of critical chaos measures for a large class of log-correlated fields. The talk is based on joint work with Janne Junnila (EPFL) and Christian Webb (Aalto University).

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Given $k$ iid $n\times n$ random Haar unitaries $U_1,\ldots , U_k$ and $l$ a fixed integer, we consider the joint behavior of $U_1^{\otimes l},\ldots , U_k^{\otimes l}$ and show that this sequence of $k$-tuples is almost surely strongly asymptotically free in the large $n$ limit. Strong asymptotic freeness is a particular case of strong convergence, which ensures the absence of outliers for a matrix model obtained from a non-commutative polynomial in the $k$-tuple  We will explain our motivations, describe some variants of our result and some applications in asymptotic representation theory. We will also elaborate on a few salient aspects of the proof, such as a theory of matrix valued non-backtracking operators, and a new inequality between moments of gaussian and unitary matrices. This talk is based on joint work in preparation with Charles Bordenave. 

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9:30–10:30am  Christopher Hoffman (University of Washington)
The shape of a random pattern-avoiding permutation

A permutation that avoids the pattern 4321 has a longest decreasing sequence of length 3. We fix n, choose \sigma a 4321-avoiding permutation uniformly at random and plot the points of the form (i/n,\sigma(i)/n) for 1 \leq i \leq n. Looking at this plot it is clear that the indices 1 through n can be partitioned into three sets. By linear interpolation from these three sets we can generate three functions. We show that the scaling limit of this measure on triples of functions is given by the eigenvalues of a ensemble of random matrices. We also discuss the scaling limits of other patterns.

10:30–11am Coffee break  

11am–12pm Gaultier Lambert (University of Zurich)
Multivariate normal approximation for traces of random unitary matrices

Let us consider a random matrix U of size n distributed according to the Haar measure on the unitary group. It is well-known that for any k≥1, Tr[U^k] converges as n tends to infinity to a Gaussian random variable and that, surprisingly, the speed of convergence is super exponential. In this talk, we revisit this problem and present non asymptotic bounds for the total variation distance between Tr[U^k] and a Gaussian. We will also consider the multivariate problem and explain how this affect the rate of convergence. We expect that our bounds are almost optimal. This is joint work with Kurt Johansson (KTH).

12–1pm Jonathan Niles-Weed (New York University)
Estimation of Wasserstein distances in the Spiked Transport Model

We propose a new statistical model, generalizing the spiked covariance model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study various probabilistic and statistical features of this model, including the estimation of the Wasserstein distance, which we show can be accomplished by an estimator which avoids the "curse of dimensionality" typically present in high-dimensional problems involving the Wasserstein distance. However, this estimator does not seem possible to compute in polynomial time, and we give evidence that any computationally efficient estimator is bound to suffer from the curse of dimensionality. Our results therefore suggest the existence of a computational-statistical gap. Joint work with Philippe Rigollet.

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Spin glasses are models of statistical mechanics where a large number of variables interact with one another, with random interactions that can be positive or negative. It is a surprisingly difficult problem to calculate the limit free energy of these models, called the Parisi formula. For a certain class of models, this problem was resolved in a remarkable series of works by Guerra-Talagrand and Panchenko. I will describe a new way to think about this result, which recasts the Parisi formula as the solution of a Hamilton-Jacobi equation. I will then explain how to use this new point of view as a proof strategy in a simpler situation related to the problem of inferring a large rank-one matrix.

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In this talk, based on joint works with Nick Cook and with Sohom Bhattacharya, I will discuss recent developments in the emerging theory of nonlinear large deviations, focusing on sharp upper tails for counts of several fixed subgraphs in a large sparse random graph, such as Erdos–Renyi or uniformly d-regular. Time permitting, I will describe our quantitative versions of the regularity and counting lemmas, which are geared for the study of sparse random graphs in the large deviations regime, and what our results suggest regarding certain questions in extremal graph theory.

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That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No prior knowledge of the subject is assumed.

 

Joint work with Ramon van Handel.

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Introduced by Chatterjee and Dembo, the nonlinear large deviation theory aims at unifying under the same paradigm certain large deviations problems such as the problem of the upper tail of sub-graph counts in Erdös-Rényi graphs, or of the traces of powers of Wigner matrices. This paradigm consists in the fact that for these large deviations problems, the optimal large deviations strategy corresponds to changes of measure which have an affine log-density with respect to the background measure. The goal of the nonlinear large deviations theory is to find a sufficient criterion for this particular strategy to be optimal in a given large deviation problem, and to propose quantitative estimates. We will discuss some improvements on this question which will lead us to develop transportation tools to prove in the case of the Gaussian measure and the uniform measure on discrete hypercube dimension-free estimates.

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We study two-dimensional random field Ising model where the external field is given by i.i.d.\ Gaussian variables with mean zero and positive variance. We show that at any non-negative temperature, the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary. This is based on joint work with Jiaming Xia.

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Derrida-Retaux model is a simple hierarchical renormalization model, originally introduced by Collet et al., that leads to many surprisingly tough questions. Some of them have been solved recently, but many others are still open. In order to answer these questions, with Yueyun Hu and Bastien Mallein, we introduced a continuous-time version of the model, which yields an exactly solvable family of solutions. We will discuss the results obtained on this model, focusing on the behavior at criticality, where a growth-fragmentation process appears as scaling limit.

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We discuss disconnection problems in two percolation models exhibiting strong correlations. These models are level sets of the Gaussian free field (GFF) and the vacant set of random interlacements, both in dimensions larger or equal to three. Specifically, we study the 'disconnection' event that either the level set of the GFF below a given level, or random interlacements, disconnect the discrete blow-up of a compact set from the boundary of an enclosing box. We give asymptotic bounds on the probability of this event in both models in their respective strongly percolative regime. Furthermore, we present results concerning the behavior of local averages of either the GFF, or occupation times of random interlacements under disconnection. In essence, the effect of disconnection amounts to a shift in the respective local average, which may be seen as an instance of entropic repulsion.  This talk is based on joint work with Alberto Chiarini.

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We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and describe the geometry of the full set of semi-infinite geodesics in a typical realization of the random environment. The main tool is the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. In the exactly solvable exponential model we give a complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics.  Part of our results concerns the existence of exceptional (random) directions in which new interesting shock structures occur.

Joint work with Chris Janjigian and Timo Seppalainen

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In the talk I will describe what is known and (mostly) unknown about asymptotic statistical topology of zero sets of random spherical harmonics of large degree on the two-dimensional sphere. I will start with basic open questions and then will discuss a non-trivial lower bound for the variance of the number of connected components of the zero set recently obtained with Fedor Nazarov. Our argument can be viewed as, probably, the first (though, modest) rigorous support of the beautiful Bogomolny-Schmit heuristics, which connects the asymptotic nodal counting with a percolation model on the square lattice.

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