Talks
Kevin Yang (Stanford)
Title: KPZ equation from ASEP with slow bond
Abstract: How does a slow bond change ASEP? At the level of SPDE limits of the height function in the weak-KPZ scaling, Franco, Goncalves, and Simon say no change. However, the methods of Franco, Goncalves, and Simon depend heavily on equilibrium tools as ASEP loses integrability with a slow bond. In this talk, we present recent work which extends (partially) the result of Franco, Goncalves, and Simon to non-stationary models through an approximate microscopic Cole-Hopf transform. This provides a step towards weak KPZ universality. We will additionally comment on other models for which a KPZ equation limit is accessible via similar methods.
Video: https://youtu.be/kLJuhPDC13s
Yuchen Liao (University of Michigan)
Title: Multi-time distribution of inhomogeneous TASEP
Abstract: Consider exponential last passage percolation with two sets of parameters where the waiting time at the (i,j)-th site is given by Exp(ai+bj). It is well-known that the last passage time G(M, N) is related to the Schur measure and has a Fredholm determinant representation. In this talk, we will discuss multi-time generalizations of these results. The main result is a formula involving multiple contour integrals of Fredholm determinants for the joint distribution of G(M1, N1), ..., G(Mk, Nk) for arbitrary many space-time points. The result is derived as a consequence of a similar formula for the related last passage percolation on a cylinder (or equivalently, TASEP on a ring) which is of interest on its own. This generalizes the recent results of Liu and Johansson-Rahman on the multi-time distribution for homogeneous geometric or exponential LPPs and the two-time distribution of an inhomogeneous geometric LPP. The crucial algebraic part is a finite sum Cauchy identity for some Grothendieck-like polynomials related to the left and right eigenvectors of the inhomogeneous TASEP generator. As a consequence, a Baik-Ben Arous-P\'ech\'e type phase transition for the large time asymptotics of height fluctuation will be described at the multi-time level.
Video: https://youtu.be/WgJjt5oWmOo
Wai-Kit Lam (UMN)
Title: Boundary and holes in first-passage percolation
Abstract: In first-passage percolation, one puts i.i.d. edge weights (t_e) on the nearest-neighbor edges on Z^d, and studies the induced (pseudo)metric. Let B(t) be the ball with radius t, centered at the origin. Of interest are the geometric properties of B(t) when t is large. In this talk, I will first go through an earlier work with Michael Damron and Jack Hanson, which studies the size of the boundary of B(t). We show that under a weak moment condition, the boundary has size of order t^{d-1} for most times. On the other hand, for heavy-tailed distributions, B(t) contains many small holes, and the boundary has a size of order strictly larger than t^{d-1}. Under the unproven uniform curvature condition on the limit shape, we show that the boundary has a size of order at most t^{d-1}(\log t)^C. I will then talk about an ongoing project with Michael Damron, Julian Gold, and Xiao Shen, in which we further study the holes in B(t). We show that for any edge weight distribution, the size of the largest hole in B(t) tends to infinity as t tends to infinity. In two dimensions, under the unproven uniform curvature assumption, we can also give an upper bound on the size of the largest hole in B(t) Title: The dynamical approach to random matrix universality
Video: https://youtu.be/h3rjkhZANTk
Benjamin Landon (MIT)
Title: The dynamical approach to random matrix universality
Abstract: We give an overview of the dynamical approach to studying the universality of the local statistics of random matrices. We introduce Dyson Brownian motion and state results on its local ergodicity, as well as how this is applied to study random matrix eigenvalues. We discuss some recent extensions of the universality theory, including the study of extremal statistics and general classes of random matrices, going beyond the Wigner class.
Video: https://youtu.be/Tu2O71_9nak
Roger Van Peski (MIT)
Title: Random matrices, random groups, singular values, and symmetric functions
Abstract: Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.
Video: https://youtu.be/CyLmrIv8Xlk
Title: qRSt: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials
Abstract: The Plancherel measure on integer partitions of size n is given by (f_lambda)^2/n!, where f_lambda is the number of standard Young tableaux of shape lambda. There are two well-known algorithms to sample partitions according to this measure. The first is to choose a permutation of n uniformly at random and apply the Robinson--Schensted (RS) correspondence. The second is to build a tableau by performing n iterations of the Greene--Nijenhuis--Wilf random hook walk, starting from the empty tableau.
In this talk, I will show that both RS and the random hook walk are degenerations of a randomized, (q,t)-dependent insertion algorithm, which we call the (q,t)-Robinson--Schensted correspondence (qRSt). This insertion algorithm gives a probabilistic proof of the equality of the coefficients of the squarefree monomial x_1...x_n y_1...y_n in the generalized Cauchy identity for Macdonald polynomials. In addition to RS and the hook walk, our algorithm degenerates to several known q- and t-deformations of RS which have appeared in recent years in connection with integrable models such as the ASEP, stochastic six vertex model, q-TASEP, and q-pushTASEP.
Video: https://youtu.be/uWw6eDeUIQo
Title: On the joint moments of characteristic polynomials of random unitary matrices
Abstract: I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.
Video: https://youtu.be/U9PwicAlbH8
Title: Temporal correlation in directed planar last passage percolation with flat initial condition.
Abstract: The directed last passage percolation on 2d lattice with exponential passage times is an exactly solvable model exhibiting KPZ growth. A topic of great interest is the coupling structure of the weights of geodesics between points as they vary in space and time. One particular case of importance being the flat initial data which corresponds to line-to-point last passage times. Answering a question asked by Ferrari and Spohn (2016), we show that for the passage times from the line x+y=0 to the points (r,r) and (n,n), their covariance is in the order of (r/n)^{4/3+o(1)}n^{2/3}, as n\to\infty and r/n being small but bounded away from zero. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are also expected to be applicable to a wider class of initial data.
This is a joint work with Riddhipratim Basu and Shirshendu Ganguly.
Video: https://youtu.be/mpoRJXlDrqw
Title: A forward-backward SDE from the 2D nonlinear stochastic heat equation
Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).
Video: https://youtu.be/y8j12kH0ibs
Title: Exit point bounds in exponential last-passage percolation and a few applications
Abstract: One versatile probabilistic approach to study directed percolation and polymer models is through comparison with their equilibrium versions when the latter are sufficiently tractable and provide a satisfactory approximation for the purposes of the problem at hand. In this talk, we focus on the paradigmatic setting of last-passage percolation with i.i.d. exponential weights on the lattice quadrant. The equilibrium versions of this model are explicitly obtained by placing additional independent exponential weights with suitable rates on the boundary (axes). Then an important aspect of the aforementioned comparison scheme is to control the point where a given geodesic from the origin exits the boundary. The main results to be presented in the talk are sharp upper bounds on the tails of the exit points.
While these bounds can be and, in part, have been concurrently established via known tail bounds for the largest eigenvalue of the Laguerre ensemble, our technique is new and relies entirely on the stationarity of the equilibrium models. We also aim to discuss two applications of the exit bounds related to the geometry of geodesics. These results provide upper bounds on the speed of distributional convergence to the Busemann limits and to the limiting direction of the competition interface.
Joint work with C. Janjigian and T. Seppäläinen.
Title: Embeddings of the Dimer model
Abstract: One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits "nice" discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. We discuss a concept of 'perfect t-embeddings' of weighted bipartite planar graphs. We believe that these embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model.
Based on: joint works with D. Chelkak, R. Kenyon, W. Lam, B. Laslier and S. Ramassamy.
Videos: https://youtu.be/Fjct-CwWvqg (Lecture 1)
https://youtu.be/Y4RfkyVCYx4 (Lecture 2)
Title: The TASEP on trees
Abstract: We study the totally asymmetric simple exclusion process (TASEP) on rooted trees. This means that particles are generated at the root and can only jump in the direction away from the root under the exclusion constraint. Our interests are two-fold. On the one hand, we study invariant measures for the TASEP on trees and provide sufficient conditions for the existence of non-trivial equilibrium distributions. On the other hand, we consider the evolution of the TASEP on trees when all sites are initially empty and study currents.
This talk is based on joint work with Nina Gantert and Nicos Georgiou.
Title: Addition of Random Matrices and Quantized Analogues
Abstract: The main objects of this talk are particle processes coming from the eigenvalues of sums of unitarily invariant random matrices and quantized analogues which arise from tensor products of irreducible representations of the unitary group. We outline an integrable probability approach to obtaining Airy point process fluctuations at the edge under an asymptotic regime where the number of summands or tensor products is sufficiently large.
Title: A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
Abstract: In this talk, I will discuss complex eigenvalues of the product of two rectangular complex Ginibre matrices that are correlated through a non-Hermiticity parameter.
In the first half, I will present the limiting spectral distribution of the model, which interpolates between classical results for random matrices on the global scale, the circular law and the Marchenko-Pastur distribution. In the second half, I will explain the microscopic behaviors of the model, which includes the limiting local correlation kernel at multi-criticality, where the interior of the spectrum splits into two connected components.
The global statistics follows from the solution of certain equilibrium measure problem and concentration for the 2D Coulomb gases on Frostman’s equilibrium measure, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.
This is based on joint work with Gernot Akemann and Nam-Gyu Kang.
Video: https://youtu.be/u1YETdOwDR8
Title: Periodic TASEP: when integrable systems meet integrable probability (once again)
Abstract: It is well-known that the Tracy-Widom distributions admit representations involving solutions to particular integrable systems. Other marginals of the KPZ fixed point, such as the Airy2 process, also admit similar representations. And very recently, first by Quastel and Remenik and shortly afterwards by Le Doussal, statistics of the KPZ fixed point were found to be connected to the KP equation.
In this talk, we plan to overview some analogue connections, but now for distributions of the periodic TASEP (pTASEP), which are believed to be the universal analogue of the KPZ universality class for periodic setup. For the step periodic initial condition, we compare the limiting one-point distribution of the pTASEP with the GUE Tracy-Widom distribution, highlighting the key features that allow to connect both of them to coupled systems of mKdV and heat equations. We also discuss some asymptotic properties of this limiting distribution, showing that it interpolates between the GUE Tracy-Widom and a Gaussian. For pTASEP with general initial condition, we also explain how very few analytic aspects of its limiting one-point distribution give a connection with the KP equation, in analogous way to Quastel-Remenik's mentioned result.
This talk is based on joint work with Jinho Baik (University of Michigan) and Zhipeng Liu (University of Kansas). Time permitting, we also briefly discuss a work in progress with Jinho Baik and Andrei Prokhorov (University of Michigan), greatly extending the mentioned results to multipoint distributions.
Video: https://youtu.be/yYLD_J7VWNI
Title: Limit theorems for descents of Mallows permutations
Abstract: The Mallows measure on the symmetric group gives a way to generate random permutations which are more likely to be sorted than not. There has been a lot of recent work to try and understand limiting properties of Mallows permutations. I'll discuss recent work on the joint distribution of descents, a statistic counting the number of "drops" in a permutation, and descents in its inverse, generalizing work of Chatterjee and Diaconis, and Vatutin. The proof is new even in the uniform case and uses Stein's method with a size-bias coupling as well as a regenerative representation of Mallows permutations.
Ofer Busani (University of Bristol)
Title: Universality of geodesic tree in last passage percolation
Abstract: In Last Passage Percolation (LPP) one assumes i.i.d. weights on the lattice Z^2. The geodesic from the anti-diagonal h(x)=-x to the point (N,N) is an up-right path starting from h and terminating at (N,N) on which the total weight is maximal. Consider now a cylinder H of width εN^2/3 and length ε^{3/2-}N centered around the point (N,N) and along the straight line going from the point (0,0) to the point (N,N). The geodesic tree consists of all the geodesics going from h and terminating in the cylinder H. We show that for exponential LPP, for a large class of weights on h(x) and with high probability, the geodesic tree coincides on H with a universal stationary tree.
Based on joint works with Marton Balazs, Timo Seppelainen and Patrik Ferrari.
Yier Lin (Columbia University)
Title: Lyapunov exponents of the SHE for general initial data
Abstract: We consider the 1+1 dimensional stochastic heat equation (SHE) with multiplicative white noise and the Cole-Hopf solution of the Kardar-Parisi-Zhang (KPZ) equation. We show an exact way of computing the Lyapunov exponents of the SHE for a large class of initial data which includes any bounded deterministic positive initial data and the stationary initial data. As a consequence, we derive exact formulas for the upper tail large deviation rate functions of the KPZ equation for general initial data. Joint work with Promit Ghosal.
Title: Matrix Means and a Novel High-dimensional Shrinkage Phenomenon.
Abstract: We analyze the impact on covariance estimation of taking a Harmonic mean as opposed to an arithmetic mean of a collection of Wishart Random Matrices in high dimensions. We see that the Harmonic mean improves on the operator norm estimation but curiously does not improve eigenvector recovery as suggested by the Davis-Kahan Inequality. Based on joint work with E. Levina and K. Levin
Title: Integrable structure behind the multitime KPZ fixed point distribution.
Abstract: We study the multitime distribution of height function of KPZ fixed point with narrow edge initial condition. The formula we use was obtained by Zhipeng Liu. It involves contour integrals of Fredholm determinant. We identify this determinant with the multicomponent KP tau function and write the system of PDE's associated with it. In particular, we generalize the result by Quastel and Remenik to the case of several times. We use the Riemann-Hilbert problem associated with the involved integrable Fredholm determinant.
Based on the joint work with Jinho Baik and Guilherme Silva.
Milind Hegde (UC Berkeley)
Title: Bootstrapping to optimal tail exponents in last passage percolation
Abstract: In planar last passage percolation (LPP), every vertex of Z^2 is given an i.i.d. non-negative weight, and the weight of directed paths between given points is maximized. In integrable models of LPP, it is well-known that the upper and lower tails of the point-to-point weight have exponents 3/2 and 3 --- matching the same for the GUE Tracy-Widom distribution. This talk will discuss a geometric route to the same exponents in non-integrable models of LPP under a set of natural assumptions on the limit shape and weak initial tail bounds, but no distributional structure of the vertex weights. Not having access to integrable formulas, the arguments rely on understanding the geometry of weight-maximizing paths, the concentration of measure phenomenon, and a natural superadditivity-based bootstrapping procedure. This is based on joint work with Shirshendu Ganguly.
Peter Nejjar (Bonn University)
Title: Dynamical Phase Transition of ASEP in the KPZ regime.
Abstract: We consider the asymmetric simple exclusion process (ASEP) on the integers. The initial data is such that in the large time density u, two rarefaction fans come together at the origin, where u jumps from 0 to 1. Placing a single second class particle at the origin initially, we show that its position converges to the difference of two independent, GUE-distributed random variables. Furthermore, we show that the limit measure of this ASEP exhibits a dynamical phase transition: When shifted on the scale of the fluctuations of the second class particle, the limit measure is a convex combination of the Dirac measure on the configuration with no particles and the Dirac measure on the configuration with no holes. The parameter in the convex combination is precisely the probability that the second class particle is to the left of the shift. Such a phase transition had been observed previously for random initial data, where the GUE-distribution is replaced by a Gaussian.
Title: Random sorting networks and last passage percolation.
Abstract: The oriented swap process is a model for a random sorting network, in which N particles, labelled 1,...,N and initially arranged in increasing order, perform adjacent swaps at random times until they reach the reverse configuration N,...,1. We establish new exact distributional identities relating the oriented swap process, the corner growth process and the last passage percolation model. One of the identities arises from the duality between the RSK and the Burge correspondences. The other one is still conjectural in its full generality and is equivalent to a new purely combinatorial identity that involves generating functions of sorting networks and Young tableaux of staircase shape, thus relating to the celebrated Edelman-Greene bijection. An application concerns the asymptotic law of the absorbing time of the oriented swap process as the number of particles gets large.
Based on joint work with Fabio Deelan Cunden, Shane Gibbons, and Dan Romik.
Erik Bates (U.C. Berkeley)
Title: Empirical measures, geodesic lengths, and a variational formula in first-passage percolation.
Abstract: We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to n e_1. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.
Duncan Dauvergne (Princeton University)
Title: Hidden invariance of last passage percolation and directed polymers.
Abstract: Planar last passage percolation and directed polymers with an integrable structure coming from either the RSK correspondence or the geometric RSK correspondence exhibit a certain Markovian structure. I will show how to combine this Markovian structure with basic translation and reflection invariance properties to produce new and interesting symmetries of these models.
Konstantin Matetski (Columbia University)
Title: Exceptional times when the KPZ fixed point violates Johansson's conjecture on maximizer uniqueness.
Abstract: In 2002, Johansson conjectured that the maximum of the Airy process minus a parabola was almost surely achieved at a unique location. This result was proved a decade later by Corwin and Hammond, as well as by other techniques in the work of Moreno Flores, Quastel, Remenik, and Pimentel. The Airy process is the fixed time spatial marginal of the KPZ fixed point run from narrow wedge initial data. It is natural to conjecture that with positive probability the KPZ fixed point spatial marginal may violate the uniqueness of the maximizer. We prove that the set of such times has Hausdorff dimension 2/3. In terms of directed polymers, these times of non-uniqueness can be thought of as times of instability in the zero temperature polymer measure, i.e. when the end point jumps from one location to another. This is a joint work in progress with I. Corwin and A. Hammond.
Title: Probabilistic conformal block for Liouville CFT on the torus.
Abstract: One of the fundamental examples of conformal field theory (CFT) is Liouville CFT which was introduced by Polyakov in the context of string theory. Recently, Liouville CFT has been rigorously constructed using probabilistic framework based on Gaussian free field and Gaussian multiplicative chaos measure. In this talk, we focus on the conformal blocks which are the key ingredients for an alternative construction of Liouville CFT pioneered by Belavin, Polyakov and Zamolodchikov. We will discuss the conformal blocks of 2d Liouville CFT on torus by using probabilistic ideas and mention a connection between our work and the AGT conjecture. This talk will be based on a joint work with Guillaume Remy, Xin Sun and Yi Sun.
Shinji Koshida (Chuo University)
Title: Pfaffian point processes from CAR algebras
Abstract: The analogy between determinantal point processes and fermionic calculi is well-known. I will give an introductory talk on this interplay from a slightly wider perspective; I will show that associated to a quasi-free state over an algebra of canonical anti-commutation relations (CAR algebra, for short), a Pfaffian point process is uniquely determined.