Guillaume BalDepartment of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027Wenjia JingDepartment of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States
Analysis of PDEsProbabilitymathscidoc:2206.03001
Discrete and Continuous Dynamical Systems, 28, (4), 1311-1343, 2010.12
We consider the theory of correctors to homogenization in stationary transport equations with rapidly oscillating, random coefficients. Let ε << 1 be the ratio of the correlation length in the random medium to the overall distance of propagation. As ε↓0, we show that the heterogeneous transport solution is well-approximated by a homogeneous transport solution. We then show that the rescaled corrector converges in (probability) distribution and weakly in the space and velocity variables, to a Gaussian process as an application of a central limit result. The latter result requires strong assumptions on the statistical structure of randomness and is proved for random processes constructed by means of a Poisson point process.
Federico CamiaDivision of Science, NYU Abu Dhabi, Saadiyat Island, Abu Dhabi, UAEJianping JiangNYU-ECNU Institute of Mathematical, Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062 P.R. CHINACharles M. NewmanCourant Institute, 251 Mercer St, New York, NY, 10012 USA
We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.
We consider Pólya urns with infinitely many colours that are of a random walk type, in two related versions. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on the offset distribution. This improves results by Bandyopadhyay and Thacker (2014–2017; convergence in probability), and Mailler and Marckert (2017; a.s. convergence assuming exponential moment).