The Probability seminar is held on Tuesday 4:05pm - 5:05pm in Ayres Hall.
The Probability seminar is held on Tuesday 4:05pm - 5:05pm in Ayres Hall.
Spring 2027
Tuesday 5/4
Tuesday 4/27
Tuesday 4/20
Tuesday 4/13
Tuesday 4/6
Tuesday 3/30
Tuesday 3/23
Tuesday 3/16: David Herzog, Iowa State University.
Title: TBA
Abstract: TBA
Tuesday 3/2: Nicole Yang, University of Tennessee Knoxville.
Title: TBA
Abstract: TBA
Fall 2026
Tuesday 12/01: Dominic Bair, University of Tennessee Knoxville.
Title: Ordered frequency in relatively hyperbolic groups.
Abstract: We study word frequencies along infinite strings whose finite prefixes lie in a regular language of geodesic representatives of group elements. Let (G, H) be a non-elementary relatively hyperbolic pair, and suppose that, over a finite symmetric generating set, there is a prefix-closed regular language containing exactly one shortest word representing each element of G. Modeling this language using a finite-state Markov chain, we define a probability measure on the space of infinite strings whose finite prefixes are words belonging to this language. Each such string labels a geodesic ray emanating from the identity. We show there is a continuous surjection from this space onto the Bowditch boundary, and the pushforward of the probability measure under this map is a Patterson--Sullivan probability obtained as a weak limit of weighted counts of group elements. For a word w, the ordered frequency of w along a given labeled ray is the limiting proportion of consecutive nonoverlapping tuples of length |w| that exactly spell w. For almost every infinite string under the constructed measure, these frequencies exist simultaneously for all nonempty geodesic words, with no restriction on their lengths. Consequently, almost every boundary point with respect to this Patterson--Sullivan probability is the endpoint of at least one such ray along which all these frequencies exist. In particular, after possibly enlarging the generating set, these results apply when the peripheral subgroups are virtually abelian.
Tuesday 11/24: Sayan Das, Vanderbilt University.
Title: TBA
Abstract: TBA
Tuesday 11/17: Xia Chen, University of Tennessee Knoxville.
Title: TBA
Abstract: TBA
Tuesday 11/10: Xia Chen, University of Tennessee Knoxville.
Title: TBA
Abstract: TBA
Tuesday 10/27: Le Chen, Auburn University.
Title: Spatial covariance of the KPZ equation from flat initial profile: one time and two times.
Abstract: We study the spatial covariance of the one-dimensional KPZ equation started from the flat initial profile, with the height function h(t,x) = log Z(t,x) given by the Cole-Hopf transform of the stochastic heat equation with constant initial data. At a fixed time t, we prove a sharp asymptotic for Cov(h(t,x), h(t,0)) as |x| tends to infinity: the covariance decays like a heat-kernel Gaussian envelope exp(-x^2/(4t)) times an explicit prefactor of order t^{3/2} |x|^{-2}. This is in sharp contrast with the narrow-wedge initial condition, where Gu and Pu recently proved a purely polynomial decay of order t/|x|. The far-field covariance of KPZ is therefore qualitatively, not just quantitatively, sensitive to the initial profile. The proof combines a Clark-Ocone representation, a boundary-layer reduction near the midpoint between the two points, and shear invariance and shear mixing of the white noise; along the way we obtain a closed-form second moment for the normalized continuum directed random polymer partition function. In the second part we pass to two times. For the flat KPZ fixed point we prove quantitative two-time spatial decorrelation: for fixed times s and t, the covariance of the heights at (t,x) and (s,0) is bounded by C exp(-c|x|^3) for |x| at least 1. Unlike the fixed-time covariance, which is governed by the Airy_1 process, the two-time covariance involves the nonlinear variational evolution of the whole earlier profile; the proof combines cubic-exponential mixing of the Airy_1 process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, centered spatial averages normalized by the square root of the window size converge to a Gaussian process whose covariance is the space-integrated two-time correlation. Based on joint work with Juan J. Jimenez (Auburn University) and with Fei Pu (Beijing Normal University).
Tuesday 10/13: Izumi Okada, University of Tokyo.
Title: Fluctuation of the structure of random walk ranges
Abstract: We investigate fluctuation phenomena for the graph distance and the effective resistance associated with random media arising from the range of a random walk. The behavior of such objects plays a central role in a variety of problems, including volume growth, metric scaling, and universality phenomena in random geometry. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to qualitatively different regimes across dimensions lower than six, equal to six, and higher than six. This is joint work with Arka Adhikari and Daisuke Shiraishi.
Tuesday 09/22: Jan Rosinski, University of Tennessee Knoxville.
Title: Distances Between Probability Distributions. Part II.
Abstract: The distances between two probability distributions (or metric) allow us to control the errors in the Central Limit Theorems, and also to investigate the rates of convergence of random elements. This talk is based on a recent paper by Solesne Bourguin and Giovanni Peccati.
Tuesday 09/15: Jan Rosinski, University of Tennessee Knoxville.
Title: Distances Between Probability Distributions.
Abstract: The distances between two probability distributions (or metric) allow us to control the errors in the Central Limit Theorems, and also to investigate the rates of convergence of random elements. This talk is based on a recent paper by Solesne Bourguin and Giovanni Peccati.
Tuesday 09/08: Barkat Mian, University of Tennessee Knoxville.
Title: A submartingale for the probability of avoiding the origin in one-point interaction ground -state diffusion: $d \in \{2,3\}$.
Abstract: We study the near-origin behavior on $[0,T]$ of the singular diffusion whose transition density is given by a Doob transform of the integral kernel of the semigroup generated by the \(d\)-dimensional Schr\"odinger operator \(L^\gamma\) with a one-point potential at the origin, where the driving family is the ground state of \(L^\gamma\) and \(d\in\{2,3\}\). We construct a submartingale whose increasing component grows only at times when the diffusion visits the origin. Using this submartingale, we show that the diffusion hits the origin with positive probability and that, conditionally on hitting the origin by time \(T\), the first hitting time has a truncated generalized inverse Gaussian (GIG) distribution. We further study the dynamics under conditioning to avoid the origin: under the conditional law, the diffusion is not a standard Brownian motion, but instead admits a representation in terms of a regularized drift and a continuous martingale. While these properties are known in dimension two, the present submartingale-based approach provides an alternative verification and treats dimensions two and three in a unified manner.
Tuesday 09/01: Barkat Mian, University of Tennessee Knoxville.
Title: Pathwise Structure of the Three-Dimensional Attractive One-Point Interaction Diffusion.
Abstract: We study the pathwise behavior of the three-dimensional attractive one-point interaction diffusion whose law was constructed by Cranston, Koralov, Molchanov, and Vainberg in [Random Oper. Stoch. Equ. 18, 73--95 (2010)], corresponding to the singular Schr\"odinger Hamiltonian
\[ \frac12\Delta+\frac{\beta}{2}\delta_0(\cdot), \qquad \beta>0. \]
We derive a local stochastic differential equation satisfied by the process away from the origin and use it to construct a natural submartingale whose increasing component in the Doob-Meyer decomposition is supported on the set of times at which the process visits the origin. In particular, we show that the process visits the origin with positive probability and that the law conditioned on avoiding the origin is three-dimensional Wiener measure. In this talk, I will also present a comparison of these results with those for the corresponding two-dimensional attractive one-point interaction diffusion, obtained in joint work with Jeremy Clark, [Electron. J. Probab. 30, 1--97 (2025)].
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