Schedule

• February 8

  Antti Knowles (ETH)

  Mesoscopic eigenvalue statistics for random band matrices

  Abstract: We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the variance and the two-point correlation function are governed by a universal power law behaviour that differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii, and describes the eigenvalue density correlations in general metallic samples with weak disorder. Our result rigorously establishes the Altshuler--Shklovskii formulas for band matrices.

• February 16

  (Unusual day, place, and time: Tuesday, ROOM 2-449, 3.00 - 4.00 pm )

  Jean-Michel Bismut (Orsay)

  Hypoelliptic Laplacian and probability

  Abstract: The hypoelliptic Laplacian is a family of operators, indexed by b ∈ R*_+ acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as b → 0 and the generator of the geodesic flow as b → ∞. The probabilistic counterpart to the hypoelliptic Laplacian is a 1-parameter family of differential equations, known as geometric Langevin equations, that interpolates between Brownian motion and the geodesic flow.

  I will present some of the probabilistic ideas that explain some of its remarkable and often hidden properties. This will include the Ito formula for the corresponding hypoelliptic diffusion, and the corresponding Malliavin calculus.

  I will also explain some of the applications of the hypoelliptic Laplacian that have been obtained so far.

• February 22

  Ewain Gwynne (MIT)

  Asymptotic behavior of the Eden model with positively homogeneous edge weights

  Abstract: The Eden model (i.e. first passage percolation with exponentially distributed edge passage times) is a random family of growing clusters in the d-dimensional integer lattice which can be defined inductively as follows. Start with a cluster consisting of the single point 0. Then, iteratively sample an edge uniformly from the set of all edges with one endpoint in the current cluster, and add the other endpoint of this edge to the cluster. We consider a generalization of the Eden model where edges are sampled with weights proportional to the values of a function f which is positively homogeneous of some real degree \alpha, rather than uniformly.

  We prove that the clusters in our model exhibits a phase transition at \alpha = 1. In particular, if \alpha < 1, then for any choice of f the clusters a.s. have a deterministic limit shape; and if \alpha > 1, then there exists an \alpha-positively homogeneous function f (which we can take to be the \alpha-th power of a norm) such that for this choice of f, the clusters are a.s. contained in a Euclidean cone of opening angle <\pi. However, we show that no single choice of norm works for every \alpha >1. This is based on a joint work with Sebastien Bubeck.

• February 29

  Pavel Bleher (IUPUI)

  The mother body phase transition in the normal random matrix model

  Abstract: We consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Ω independent of the cut-off, and we determine the domain Ω explicitly by finding a rational parametrization of its boundary.

  We study in detail the mother body problem associated to Ω. It turns out that the mother body measure μ* displays a novel phase transition, which we call the mother body phase transition: although ∂ Ω evolves analytically, the mother body measure μ* undergoes a `one-cut to three-cut' phase transition.

  We consider multiple orthogonal polynomials associated to the normal matrix model. Developing the Deift-Zhou nonlinear steepest descent method to the associated Riemann--Hilbert problem, we obtain strong asymptotic formulas for these polynomials, and we prove that the distribution of their zeros converges to the mother body measure μ*.

  This is a joint project with Guilherme Silva.

• March 7

  Yu-Ting Chen (Harvard)

  Non-uniqueness in SPDEs with non-Lipschitz noise coefficients

  Abstract: One problem for uniqueness in SPDEs, open for more than two decades, is to determine pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion for nonnegative solutions. A positive result would establish an infinitedimensional analogue of the celebrated Yamada-Watanabe sharp uniqueness for SDEs with non-Lipschitz noise coefficients. However, a recent work by Mueller, Mytnik and Perkins proves that, among other things, dropping the assumption of nonnegative solutions leads to pathwise non-uniqueness. This result leaves open the question whether reflection of nonnegative solutions around their zeros would be a mechanism sufficient for the uniqueness.

  In this talk, I will discuss a result which continues the investigation of the SPDE of super-Brownian motion by Mueller et al. I will first review the SPDE of superBrownian motion and related aspects. Then I will introduce certain perturbations for the SPDEs, and discuss a non-uniqueness result for the nonnegative solutions of these perturbed SPDEs.

• March 14

  Antonio Auffinger (Northwestern)

  The Parisi Formula: duality and equivalence of ensembles

  Abstract: In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer, called the Parisi measure. This remarkable formula was proven by Talagrand in 2006. In this talk I will explain a new representation of the Parisi functional that finally connects the temperature parameter and the Parisi measure as dual parameters.

  Based on joint-works with Wei-Kuo Chen.

• March 21

  Spring Vacation

• March 28

  Mykhaylo Shkolnikov (Princeton)

  Edge of beta ensembles and the stochastic Airy semigroup

  Abstract: Beta ensembles arise naturally in random matrix theory as a family of point processes, indexed by a parameter beta, which interpolates between the eigenvalue processes of the Gaussian orthogonal, unitary and symplectic ensembles (GOE, GUE and GSE). It is known that, under appropriate scaling, the locations of the rightmost points in a beta ensemble converge to the so-called Airy(beta) process. However, very little information is available on the Airy(beta) process except when beta=2 (the GUE case). I will explain how one can write a distribution-determining family of observables for the Airy(beta) process in terms of a Brownian excursion and a Brownian motion. Along the way, I will introduce the semigroup generated by the stochastic Airy operator of Ramirez, Rider and Virag. Based on joint work with Vadim Gorin.

• April 4

  Jeremy England (MIT)

  Emergent Fine-tuning to Environmental Drives in a Random Chemical Mixture

  Abstract: The steady-state behavior of an undriven mixture of reaction chemical species is the equilibrium point where the concentrations obey a simple exponential relationship to free energy. Once external environmental drives are introduced, however, steady-state concentrations may deviate from these equilibrium values via processes that require sustained absorption and dissipation of work. From a physical standpoint, the living cell is a particularly intriguing example of such a nonequilibrium system because the environmental work sources that power it are relatively difficult to access -- only the proper orchestration of many distinct catalytic actors leads to a collective behavior that is competent to harvest and exploit available metabolites. Here, we study the dynamics of an in silico chemical network with random connectivity in a driving environment that only makes strong chemical forcing available to rare combinations of concentrations of different molecular species. We find that the long-time dynamics of such systems are typified by the spontaneous extremization of forcing, so that the molecular composition converges on states that exhibit exquisite fine-tuning to available work sources.

• April 11

  Jinho Baik (Michigan)

  Fluctuations of the free energy of spherical spin glass

  Abstract: We consider the spherical spin glass model (the spherical Sherrington-Kirkpatrick model). We evaluate the fluctuations of the free energy when there are only 2-spin interactions, and show that the limit is given by the GOE Tracy-Widom distribution for low temperature and by the Gaussian distribution for high temperature. Non-Gaussian interactions and non-zero mean interactions are also going to be discussed. This is a joint work with Ji Oon Li.

• April 18

  Patriots Day

• April 25

  Marek Biskup (UCLA)

  Conformal symmetries of the extremal process associated with 2D Discrete Gaussian Free Field

  Abstract: The 2D Discrete Gaussian Free Field (DGFF) is a mean-zero Gaussian process defined on a finite subset of the square lattice with covariance given by the Green function of the simple symmetric random walk killed upon exit from the set. This field received considerable attention due to the fact that it exhibits many interesting asymptotic properties and, in particular, has a conformally invariant scaling limit. In my talk I will focus on the extreme values of the DGFF and show how conformal invariance reflects itself in the scaling limit thereof. Based on a sequence of joint papers with Oren Louidor.

• May 2

  Leonid Petrov (UVA)

  The quantum integrable particle system on the line

  Abstract: I will discuss the higher spin six vertex model - an interacting particle system on the discrete 1d line in the Kardar--Parisi--Zhang universality class. Observables of this system admit explicit contour integral expressions which degenerate to many known formulas of such type for other integrable systems on the line in the KPZ class, including stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. The structure of the higher spin six vertex model (leading to contour integral formulas for observables) is based on Cauchy summation identities for certain symmetric rational functions, which in turn can be traced back to the sl(2) Yang--Baxter equation. This framework allows to also include space and spin inhomogeneities into the picture, which leads to new particle systems with unusual phase transitions.

• May 9

  Ivan Corwin (Columbia)

  Dynamics preserving two-dimensional Gibbs ensembles and Macdonald processes

  Abstract: We study a class of Gibbs measures on two-dimensional arrays of particles which can be thought of as q-deformations of the uniform measure on interlacing configurations, and introduce certain simple local Markov dynamics which preserves this class of measures. From this surprising structure we deduce many results including Gaussian free field / additive stochastic heat equation fluctuations for certain limits of these measures / dynamics, and regularity properties for the Kardar-Parisi-Zhang equation. Other objects we related to this structure includes Gibbs measures on tilings / dimers, the O'Connell-Yor and log-gamma directed polymers, Dyson Brownian motion, the KPZ / Airy line ensemble and the O'Connell-Warren multilayer extension of the stochastic heat equation.