Stochastics Seminar

Many classical results for the Brownian motion rely on its special properties such as independent increments and Markov property. When moving away from the Brownian motion, strong local nondeterminism (SLND) becomes a useful tool for studying Gaussian random fields including solutions to stochastic PDEs (SPDEs) with additive Gaussian noise. In this talk, I will present some results on SLND of SPDEs and their applications.

Symmetries such as self-similarity are fundamental concepts that appear across mathematics, from probability and operator theory to mathematical physics. In this talk, I will explore how a novel and constructive algebraic perspective can deepen our understanding of these phenomena. I will begin by discussing how an alternative approach to the renormalization group method provides new insights into scaling limits and universality. I will proceed by presenting a unified framework for understanding self-similarity and Lie symmetries through the lens of group representation theory, operator algebras and spectral theory. This approach reveals the central role of the spectral representation in the Hilbert space of stochastic objects and offers a constructive strategy rooted in the celebrated Stone-Von Neumann-Mackey theorem. Within this framework, I will highlight the surprising roles of the Bessel operator and the Laplacian, which provide unexpected connections and insights.

Two core objects in the KPZ universality class are the KPZ fixed point (KPZFP), a Markov process with state space consisting of upper semi-continuous functions, and the directed landscape (DL). These serve as the scaling limits of random growth processes and random planar geometry, respectively. I will talk about a recent work with Duncan Dauvergne, in which we establish an unexpected new characterization of the DL as the unique natural random field driving the KPZFP. This effectively provides a new construction of the DL from the KPZFP without relying on exactly solvable structures. Building on this, we further establish convergence to the DL for a range of models, including some that lack exact solvability, such as general 1D exclusion processes, various couplings of ASEPs (e.g., the colored ASEP), the Brownian web and random walk web distances, and directed polymers.

I shall discuss first passage percolation on Cayley graphs of Gromov hyperbolic groups under mild conditions on the passage time distribution. Appealing to deep geometric and topological facts about hyperbolic groups and their boundaries, several questions become more tractable in this set up compared to their counterparts in Euclidean lattices. In particular, i shall describe several results about time constants, fluctuations, coalescence of geodesics, and exceptional directions where coalescence fails. Some of the results are parallel to what are expected in Euclidean background geometry, while substantially different features are exhibited in other aspects. Based on joint works with Mahan Mj.

Liouville field theory is the model for two-dimensional quantum gravity. Recently, probabilistic methods were applied to it and achieved significant success. The main contributions were the definition of the Polyakov integral analog, proof of conformal bootstrap, and the DOZZ formula for three-point correlation functions on the sphere. The theory was developed not only on the Riemann sphere but also on Riemann surfaces. Our work focuses on the case of the torus. The conformal bootstrap formula expresses one point correlation function in terms of conformal blocks. Traditionally, they are described using asymptotic series. The probabilistic methods provided expectation type formula for them and allowed to show that asymptotic series is converging in a small disc. Liouville field theory has central charge c associated with it. Zamolodchikov in 1984 conjectured the description of conformal blocks as c goes to infinity. The corresponding limiting object was called semi-classical conformal blocks. We use a probabilistic formula for conformal blocks to prove the Zamolodchikov conjecture and show that asymptotic series for semi-classical conformal blocks is converging in small disc. This is a joint work with Harini Desiraju and Promit Ghosal.

Stationary measures are well-understood for various models in the KPZ universality class; for instance, Brownian motion is stationary for the KPZ equation, and a random walk is stationary for the log-gamma polymer. For models on a half-space, however, stationary measures are more complicated and have only recently been described. I will discuss a result for the half-space log-gamma polymer establishing convergence (i.e., attraction) to a certain stationary measure along the anti-diagonal path, which can be seen as the top layer of a pair of random walks “softly” conditioned never to intersect one another. Our proof takes as a starting point the recently constructed half-space log-gamma line ensemble (Barraquand—Corwin—Das), and I will describe a method of leveraging the Gibbs line ensemble structure to establish such a convergence result. This is based on joint work with Sayan Das.

For critical percolation on the 2D triangular lattice, consider the probability that three points lie in the same cluster. The Delfino-Viti conjecture predicts that in the fine mesh limit, under suitable normalization, this probability converges to the imaginary DOZZ formula from conformal field theory. We prove the Delfino-Viti conjecture, and more generally, obtain the cluster connectivity three-point function of the conformal loop ensemble. Our arguments depend on the coupling between Liouville quantum gravity and the conformal loop ensemble. Based on joint work with Gefei Cai, Xin Sun, and Baojun Wu.

In this work, we develop a formal system of inductive logic. It uses an infinitary language that allows for countable conjunctions and disjunctions. It is based on a set of nine syntactic rules of inductive inference, and contains classical first-order logic as a special case. We also provide natural, probabilistic semantics, and prove both $\sigma$-compactness and completeness. We show that the whole of modern, measure-theoretic probability theory is properly embedded in this system of inductive logic. The semantic models of inductive logic are probability measures on sets of structures. (Structures are the semantic models of finitary, deductive logic.) Moreover, any probability space, together with a set of its random variables, can be mapped to such a model in a way that gives each outcome, event, and random variable a logical interpretation. This embedding, however, is proper. There are probabilistic ideas that are expressible in this system of logic which cannot be formulated in a classical measure-theoretic probability model. One such idea is the principle of indifference, a heuristic notion originating with Laplace. Roughly speaking, it says that if we are ``equally ignorant'' about two possibilities, then we should assign them the same probability. The principle of indifference has no rigorous formulation in modern probability theory. It exists only as a heuristic. Moreover, its use has a history of being problematic and prone to apparent paradoxes. Within inductive logic, however, we provide a rigorous formulation of this principle, and illustrate its use through a number of typical examples. Many of the ideas in inductive logic have counterparts in measure theory. The principle of indifference, however, does not. Its formulation requires the structure of inductive logic, both its syntactic structure and the semantic structures embedded in its models. As such, it exemplifies the fact that inductive logic is a strictly broader theory of probability than any that is based on measure theory alone.