Probability and Statistics Seminar
Spring 2020
The seminars will be held online using Zoom. The Zoom address will be emailed to those who are interested.
Meeting Times: Wednesdays 4pm- 5pm
Please contact Zhipeng Liu for arrangements.
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Date | |
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April 15 |
David Nualart (University of Kansas) Using techniques of Malliavin calculus combined with Stein’s method for normal approximations we derive a central limit theorem for spatial averages of the solution to the parabolic Anderson model with delta initial condition. |
April 22 |
Xin Sun (Columbia University) Conformal geometry of random surfaces in 2D quantum gravity From a probabilistic perspective, 2D quantum gravity is the study of natural probability measures on the space of all possible geometries on a topological surface. One natural approach is to take scaling limits of discrete random surfaces. Another approach, known as Liouville quantum gravity (LQG), is via a direct description of the random metric under its conformal coordinate. In this talk, we review both approaches, featuring a joint work with N. Holden proving that uniformly sampled triangulations converge to the so called pure LQG under a certain discrete conformal embedding. |
April 29 |
Lingfu Zhang (Princeton University) Temporal correlation in last passage percolation with flat initial condition 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 are varied in space and time. One particular case of importance being the flat initial data which corresponds to line-to-point last passage times. Settling a conjecture 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 Θ((r/n)^{4/3+o(1)}n^{2/3}), as n→∞ 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 for a wider class of initial data. This is a on joint work with Riddhipratim Basu and Shirshendu Ganguly. |
May 6 |
Jin Feng (University of Kansas) Hydrodynamic limit large deviation for stochastic Carleman particles, a Hamilton-Jacobi approach The deterministic Carleman equation can be considered as an one dimensional two speed fictitious gas model. Its associated hydrodynamic limit gives a nonlinear heat equation. The first rigorous derivation of such limit was given by Kurtz in 1973. In this talk, starting from a more refined stochastic model giving the Carleman equation as mean field, we derive a macroscopic fluctuation/large deviation structure associated with the hydrodynamic limit. The large deviation is established through an abstract Hamilton-Jacobi method applied to this specific setting, which involves making sense of Hamilton-Jacobi partial differential equations in the space of measures. After an initial linkage with such abstract equations, we start a probability-free approach to derive the probabilistic large deviation result. The principal idea is to identify a two scale averaging structure in the context of Hamiltonian convergence in the space of probability measures. This is conceptually achieved through a change of coordinate to the density-flux description of the problem and extending a method in the weak KAM theory to the infinite particle context (for explicitly identifying the effective Hamiltonian. This is joint work with Toshio Mikami and Johannes Zimmer. While we rely on a very probabilistic model and derive a probabilistic result, the broader goal is about testing potentially useful new ideas for deriving deterministic hydrodynamics. |
May 13 |
Maurizia Rossi (University of Milano-Bicocca) Nodal lengths of random spherical harmonics In this talk we deal with the geometry of (random) Laplacian eigenfunctions [Ber77, CM18, Log18, MPRW16, NPR19, Yau82]. In particular, we investigate the asymptotic distribution, in the high-energy limit, of the nodal length for random spherical harmonics [MRW20]. Moreover, we study the correlation between the latter and the measure of the boundary for excursion sets at any non-zero level [MR19]. References [Ber77] M.-V. Berry. “Regular and irregular semiclassical wavefunctions”. Journal of Physics A: Mathematical and Theoretical 10, 12:2083–2091, 1977. [CM18] V. Cammarota and D. Marinucci. “A quantitative central limit theorem for the Euler–Poincar ́e characteristic of random spherical eigenfunctions”. Annals of Probability, 46(6): 3188–3228, 2018. [Log18] A. Logunov. “Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture”. Annals of Mathematics, 187(1):241–262, 2018. [MPRW16] D. Marinucci, G. Peccati, M. Rossi, and I. Wigman. “Non-Universality of nodal lengths distribution for arithmetic random waves”. Geometric and Functional Analysis, 26(3):926–960, 2016. [MR19] D. Marinucci and M. Rossi. “On the correlation between nodal and boundary lengths for random spherical harmonics”. Preprint arXiv:1902.05750. [MRW20] D. Marinucci, M. Rossi and I. Wigman. “The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics”. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 56, 1, 374–390 (2020). [NPR19] I. Nourdin, G. Peccati and M. Rossi. “Nodal statistics of planar random waves”. Communications in Mathematical Physics, 369(1): 99–151, 2019. [Yau82] S.-T. Yau: “Survey on partial differential equations in differential geometry”. Seminar on Differential Geometry, volume 102 of Annals of Mathematical Studies, pages 3–71. Princeton University Press, Princeton, N.J., 1982. |