We study the point process limits of the circular Jacobi beta-ensemble and the real orthogonal beta-ensemble. Using the differential operator framework introduced by Valkó and Virág, we identify the point process scaling limits as spectra of certain stochastic differential operators. This framework also allows us to prove the convergence of the normalized characteristic polynomials of the finite models to certain random analytic functions. In this talk, I will first review the theoretic operator framework and these constructions. Then I will present the operator level convergence and several characterizations of the limiting objects.

This talk is based on joint work with Benedek Valkó.

We establish tail estimates for the averaged empirical distribution of edge weights along geodesics in first-passage percolation on Z^d. Our upper bounds also come with exponentially decaying tail estimates for the empirical distribution. As an application, we show that even for an edge weight distribution with no positive moment, the limit point of the averaged empirical distribution of finite geodesics could have finite positive moments of all orders. (Joint work with Michael Damron, Christopher Janjigian and Wai-Kit Lam.)

In this talk, I first set the framework and briefly mention recent works on central limit theorems for solution of certain SPDEs. Then I will focus on a hyperbolic Anderson model driven by a Gaussian homogeneous noise that is colored in time and space. We will see that the spatial average of the solution admits Gaussian fluctuation and the rate of convergence can be characterized by the total-variation distance. The main difficulty of the proof lies in the fact that the driving noise is not white, so that Ito techniques are no more useful in this case. This talk is based on joint works with R. Balan, D. Nualart and L. Quer-Sadanyons (arXiv: 2101.10957) and D. Nualart and P.Q. Xia (arXiv: 2109.03875).

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\xi], where \xi is a fixed unit vector. 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.

The KPZ universality class is believed to contain a very broad collection of models of stochastic growth. A theme in KPZ that has developed a great deal over the last 15 years, and particularly in recent years, is the presence of Brownian behaviour---the classical kind of universality---in many natural objects. In this talk I will survey some of the results in the zero-temperature setting---concerning objects such as last passage percolation, the Airy_2 process, and the KPZ fixed point---focusing on recent advances in obtaining and applying quantitative process-level Brownian comparisons of the Airy_2 process, as well as connections to the behaviour of geodesics. Based on joint work with Jacob Calvert, Ivan Corwin, Alan Hammond, and Konstantin Matetski.

We consider the generalized Langevin equation (GLE) describing motions of particles in viscoelastic fluids. In the first part of the talk, we discuss anomalous diffusion of the GLE when the particle is either moving freely or trapped in harmonic potentials. We characterize the large-time asymptotics of the mean-squared displacements in terms of the power-law decay of the memory. In the second part of the talk, we discuss the GLE in non-linear potentials. Using a Gibbsian framework, we provide conditions on the decay of the memory to ensure uniqueness of statistically steady states. This generalizes previous known results for the GLE under particular kernels as a sum of exponentials.

We will present recent results on developing new deep learning algorithms leveraging differential equations and random graph insights. First, we will present a new class of continuous-depth deep neural networks that were motivated by the ODE limit of the classical momentum method, named heavy-ball neural ODEs (HBNODEs). HBNODEs enjoy two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly accelerate learning and improve the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data. Second, we will extend HBNODE to graph learning leveraging diffusion on graphs, resulting in new algorithms for deep graph learning. The new algorithms are more accurate than existing deep graph learning algorithms and more scalable to deep architectures, and also suitable for learning at low labeling rate regimes. Moreover, we will present a fast multipole method-based efficient attention mechanism for modeling graph nodes interactions. Third, if time permits, we will discuss building an efficient and reliable overlay network for decentralized federated learning based on the random graph theory.

The long-run variance matrix and its inverse, the so-called precision matrix, give, respectively, information about correlations and partial correlations between dependent component series of multivariate time series around zero frequency. This talk will present non-asymptotic theory for estimation of the long-run variance and precision matrices for high-dimensional Gaussian time series under general assumptions on the dependence structure including long-range dependence. The presented results for thresholding and penalizing versions of the classical local Whittle estimator ensure consistent estimation in a possibly high-dimensional regime. The highlight of this talk is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model.

The coupling (hydrodynamical) approach to study the Kardar-Parisi-Zhang (KPZ) fluctuations in stochastic planar models originated in the works of E. Cator and P. Groeneboom on Hammersley's process and the Poisson last-passage percolation (LPP) around 2005. Based on invariant measures, couplings and comparison arguments, it was conceived as a more classical probabilistic alternative to the sophisticated techniques of integrable probability drawing from representation theory, combinatorics and asymptotic analysis. Thanks to the contributions of multiple researchers (including M. Balazs, E. Cator, P. Groeneboom, P. A. Ferrari, J. Martin, L. Pimentel, F. Rassoul-Agha, T. Seppalainen, B. Valko among others), the coupling method has been systematically developed over the years to treat many aspects of the KPZ universality in various models of directed LPP, directed polymers and interacting particles. The purpose of this talk is to present some technical advances within the coupling framework, which are powered by a very recently discovered connection to certain m.g.f. identities of E. Rains from 2000. These identities elevate the capabilities of the coupling approach to the level of integrable probability for many interesting problems. As a concrete example, we focus on a new coupling derivation of optimal-order central moment bounds in one of the paradigmatic KPZ-class models, the exponential LPP. Based on (partly ongoing) joint works with N. Georgiou, C. Janjigian, J. Ortmann, T. Seppalainen and Y. Xie.

Hermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We introduce new representations of Hermite processes with multiple Wiener-Ito integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. We shall also introduce new limit theorems which correspond to the physical meaning of the representation.

Tail and exit point estimates in models within the Kardar-Pairis-Zhang universality class have played a particularly important role in mathematical work seeking to make physically motivated heuristic arguments about random growth models rigorous. Previous approaches to proving these estimates in integrable growth models have centered on case-by-case analysis using inputs from integrable probability. This talk will discuss a new probabilistic approach based on coupling which gives a unified argument proving sharp upper-tail moderate deviation and exit point bounds for all known integrable directed polymer models simultaneously. Based on joint works (in progress) with Elnur Emrah, Timo Seppalainen, and Yongjia Xie.

In exponential directed last-passage percolation, each vertex in Z^2 is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. It is a natural question to ask what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present some new results on this. We show convergence of the distribution of the 'environment' as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure, as the geodesic length goes to infinity. In addition, we obtain an explicit description of the limiting environment, which depends on the direction of the geodesic. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples. This is based on joint work with James Martin and Allan Sly.