Probability and Mathematical Physics Seminar

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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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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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• 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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• 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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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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