Date/Time : 31 Jul 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jan M. Swart, The Czech Academy of Sciences

In a classical game two players, Alice and Bob, take turns to play n moves each. Alice starts. For each move each player has two options, 1 and 2. The outcome is determined by the exact sequences of moves played by each player. Prior to the game, a winner is assigned to each of the 2^{2n} possible outcomes in an i.i.d. fashion, where p is the probability that Bob is the winner for a given outcome. Then it is known that there exists a value 0<p_c<1 such that the probability that Bob has a winning strategy for large n tends to one if p>p_c and to zero if p<p_c. We study a modification of this game for which the outcome is determined by the exact sequence of moves played by Alice as before, but in the case of Bob all that matters is how often he has played move 1. We show that also in this case, there exists a sharp threshold p’_c that determines which player has with large probability a winning strategy in the limit as n tends to infinity.

Joint work with Anja Sturm and Natalia Cardona-Tob\’on (G\”ottingen).

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Date/Time : 12 Jul 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Nishant Chandgotia, TIFR CAM (Bangalore)

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In these series of talks I will describe some recent progress in 3 dimensions where we show a large deviations principle for dimer tilings analogous to those by Cohn, Kenyon and Propp (2000). This is joint work with Scott Sheffield and Catherine Wolfram.

The talks should be accessible to a general audience but it will help to have working knowledge of probability and basic ergodic theory.

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Date/Time : 10 Jul 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Nishant Chandgotia, TIFR CAM (Bangalore)

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In these series of talks I will describe some recent progress in 3 dimensions where we show a large deviations principle for dimer tilings analogous to those by Cohn, Kenyon and Propp (2000). This is joint work with Scott Sheffield and Catherine Wolfram.

The talks should be accessible to a general audience but it will help to have working knowledge of probability and basic ergodic theory.

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Date/Time : 18 Apr 2024 16:30
Venue : S14 #06-19
Speaker : Ching Wei Ho, Academia Sinica

It is a classical problem to study the evolution of roots of polynomials under application of a differential operator. In this talk, I will discuss the heat evolution of random polynomials with a rotationally invariant root distribution on the complex plane. The limiting root distribution of the heat-evolved random polynomial can be completely determined in terms of its log potential. For example, when a Weyl polynomial, whose root distribution converges to the uniform distribution on the unit disk, undergoes heat flow, the limiting root distribution is uniform on some ellipse until time 1 at which it becomes exactly the semicircle law. This is joint work with Brian Hall, Jonas Jalowy, and Zakhar Kabluchko.

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Date/Time : 18 Mar 2024 16:00
Venue : S16 #03-04
Speaker : Stefan Junk, Gakushuin University

We consider the directed polymer model, which describes paths affected by a random space-time environment in spatial dimension d>2. The model undergoes a phase transition between a high-temperature, weak disorder phase and a low-temperature, strong disorder phase, which is characterized by whether the associated (normalized) partition function converges to zero. From the physical point of view, it is more natural to consider a very-strong-disorder regime, characterized by whether the partition function converges to zero exponentially fast. It has been a long-standing conjecture that these two notions are equivalent and we now give a proof of this. Moreover, our proof reveals that weak disorder holds at the critical value. Joint work with Hubert Lacoin.

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Date/Time : 13 Mar 2024 16:00
Venue : S16 #03-09
Speaker : Hampus Nyberg, KTH Stockholm

Deep zero problems consider holomorphic functions satisfying local properties at a small number of points. We take a look at one such problem about uniqueness in the Bargmann-Fock space and show that it reduces down to a question about products such as $\frac{\sin{1}}{\sin{2}}\frac{\sin{3}}{\sin{4}}\frac{\sin{5}}{\sin{6}}…$.

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Date/Time : 25 Jan 2024 15:00
Venue : S16-06-118, Seminar Room
Speaker : Zhou Fan, Yale University

In statistical and probabilistic models, “mean-field” behavior arises when the aggregate effect of many variables is well-approximated by an average or effective field. Ideas from the analysis of mean-field models in statistical physics have been influential in our understanding of various problems of high-dimensional statistics in recent years. In this talk, I will argue that statistical applications can conversely drive the basic study of mean-field phenomena.

As a motivating example, I will discuss an application of high-dimensional mean-field theory to a nonparametric empirical Bayes procedure for PCA. I will then describe several developments of the theory beyond i.i.d. models, including extensions of Approximate Message Passing algorithms and analyses of spin glass models where the couplings matrices are orthogonally invariant in law.

Joint work w/ Yufan Li, Subhabrata Sen, Chang Su, Tianhao Wang, Yihong Wu, Xinyi Zhong.

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Date/Time : 17 Aug 2023 10:30
Venue : S17 #05-11 (SR5)
Speaker : Sayan Das, Columbia University

Sticky Brownian Motions are a family of correlated Brownian motions that have a tendency to stick together. They can be realized as the scaling limit of random walks in random environments and can also be viewed as random motions in a continuum-random environment. In this talk, I will present some results related to the quenched density of the motion of a particle in this continuum-random environment. In particular, I will show that under moderate deviation regime (a particular regime in between large deviation regime and diffusive regime), this quenched density after rescaling weakly converges to the solution of the stochastic heat equation with multiplicative space-time white noise. I will explain how our results shed light on the extreme behavior of Sticky Brownian Motions. This is a joint work with Hindy Drillick and Shalin Parekh.

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Date/Time : 18 Apr 2023 11:00
Venue : S17 #04-05 (SR2)
Speaker : Simon Gabriel, University of Warwick

We study the Allen-Cahn equation, commonly used to describe convection phenomena in physics. It is natural to start the system from chaos, corresponding to an initial condition described by spatial white noise. In two spatial dimensions, this leads to a critical initial datum, for which the equation can not be made sense of a priori.

In this talk we give a gentle introduction into the methods used to study this equation using an infinite expansion of stochastic integrals. Both the convergence and truthfulness of such an approximation are non-trivial, since the criticality has the effect that each term in the infinite expansion contributes towards the solution. Thus, it is crucial to determine each term’s statistic exactly, for which we exploit a tree-like notation. This allows us to manipulate stochastic integrals using solely their underlying tree structure. After performing such manipulations, we can conclude convergence of the expansion to a solution with the help of a series expansion of an explicit ODE.

The talk is based on joint work with Tommaso Rosati and Nikos Zygouras.

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