Purdue Probability Seminar, Spring 2022

In this talk, I will report a recent joint work with Raluca Balan and Xia Chen [BCC22]. In this work, we study the stochastic wave equation in dimensions \(d\leq 3\), driven by a Gaussian noise \(\dot{W}\) which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the \(p\)-th moment of the solution when either the time or \(p\) goes to infinity. For the critical case, namely, when \(d=3\) and the noise is white, we obtain the exact transition time for the second moment to be finite. The main obstacle for this work is the lack of the Feynman-Kac representation for the moment, which has been overcome by a careful analysis of the Wiener chaos expansion. Our methods turn out to be very general and can be applied to a broad class of SPDEs. When time permits, I will show the interpolation results with my Ph.D. student N. Eisenberg [CE22] where both stochastic heat and wave equations can be studied in the same framework.

References:
[BCC22] Balan, Raluca M., Chen, Le, and Chen, Xia. "Exact asymptotics of the stochastic wave equation with time-independent noise." Ann. Inst. Henri Poincaré Probab. Stat, to appear. Preprint at arXiv:2007.10203.
[CE22] Chen, Le and Eisenberg, Nicholas. "Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics." Stoch. Partial Differ. Equ. Anal. Comput. (pending revision) preprint at arXiv:2108.11473.

In this talk, I will give an elementary introduction to the Bethe Ansatz, invented by Hans Bethe in 1931. This is a mathematical framework to find eigenvalues and eigenvectors of certain Hamiltonians describing one-dimensional spin chains analytically. I will discuss two examples, the asymmetric XXZ quantum spin chain and the two-species asymmetric diffusion model, to explain the general idea of the Bethe Ansatz and give some insight into methods of calculating solutions.

We investigate a self-interacting random walk, in a dynamically evolving environment, which is a random tree built by the walker itself, as it walks around. At time \(n=1,2,\dots\), right before stepping, the walker adds a random number (possibly zero) \(Z_n\) of leaves to its current position. We assume that the \(Z_n\)'s are independent but we do not assume that they are identically distributed. We obtain non-trivial conditions on their distributions under which the random walk is recurrent. This result is in contrast with previous work in which, under a sort of uniform ellipticity condition, namely, that \(\inf_n P(Z_n\geq 1)=\kappa>0\), the random walk was shown to be ballistic. We also obtain results on the transience of the walk, and the possibility that it ``gets stuck.'' From the perspective of the environment, we provide structural information about the sequence of random trees generated by the model when \(Z_n\sim \mathsf{Ber}(p_n)\), with \(p_n=\Theta(n^{-\gamma})\) and \(\gamma \in (2/3,1].\) We prove that the empirical degree distribution of this random tree sequence converges almost surely to a power-law distribution of exponent \(3\), thus revealing a connection to the well-known preferential attachment model. This is joint work with R. Ribeiro (Boulder) and G. Iacobelli (Rio de Janeiro).

We will discuss stochastic quantization of the Yang-Mills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the Yang-Mills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also briefly describe the construction of this orbit space, on which we have well-defined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer.

We establish upper and lower bounds for the size of the largest hole in the infected region \(B(t)\) in first-passage percolation (FPP). The lower bound ``\(\log (t)\)" holds for general edge weight \(\tau_e\) with the minimum assumptions that \(\tau_e\) is non-deterministic and satisfies \(\mathbb{P}(\tau_e = 0) < p_c(d)\). The upper bound ``\(\log(t)^C\)" holds for planner FPP under the additional assumptions that \(\tau_e\) has an exponential moment and the uniform curvature condition holds for the limit shape of \(B(t)\). (Joint work with Michael Damron, Julian Gold, Wai-Kit Lam.)

We will present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in the Euclidian space. The dynamics we study are those of a Markov birth-death process. We prove functional limit theorems in the so-called thermodynamic regime using the recent theory of Malliavin-Stein bounds. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs. (Joint work with Omer Bobrowski and Robert J. Adler)

Erdös-Rényi random graphs with constant average degree are interesting due to their similarity to real-world networks. Unfortunately, because they are very sparse, many classical tools that were developed for denser graphs stop working. Consequently, despite the recent progress in understanding large deviation properties of non-linear observables of random graphs, most questions in this regime of sparsity remain open. In this talk I will present two recent results in this direction, based on joint work with Shirshendu Ganguly and Kyeongsik Nam, and relying on an analysis of the geometry of the graph that is specific to the problem at hand. The first question is about the upper tail behavior of the number of triangles, a problem that had remained open for a couple of decades. Secondly I will talk about large deviations of the largest eigenvalue of weighted graphs, in particular about the effect of the distribution of the edge weights, and present a surprising universality result.

The probabilistic coupling 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. It has since been developed 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 recently discovered connection to certain m.g.f. identities of E. Rains from 2000. As a concrete example, we focus on a new coupling derivation of optimal-order central moment bounds in one of the representative KPZ-class models, the exponential LPP.

Based on (partly ongoing) joint works with N. Georgiou, C. Janjigian, J. Ortmann, T. Seppäläinenn and Y. Xie.

In the last passage percolation models (predicted to lie in the KPZ universality class), geodesics are oriented paths moving through random noise, accruing maximum weight. The weights of such geodesics (as their endpoints vary) give rise to a random energy field. It is expected to converge to a universal object known as the Directed Landscape, which satisfies various scaling invariance properties, making it a source of intricate random fractal behaviors. I will talk about recent results in this direction, on the coupling structure of the geodesic weights as the endpoints vary. The main result is that we find that the non-constancy set of the difference profile has Hausdorff dimension 2/3, in the temporal direction. This requires several new ideas, compared to previous results of this set in the spatial direction. A key ingredient is the construction of the local time process for the geodesic, akin to Brownian local time, and showing that the local time process has Hausdorff dimension 1/3. This is joint work with Shirshendu Ganguly.

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction. In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.