### Random conductance models with stable-like long range jumps

Abatract

In this talk, we consider random conductance models with stable-like long range jumps, and obtain the quenched invariance principle (QIP) and a quantitative version of stochastic homogenization for the scaled random walks with explicit polynomial rates up to logarithmic corrections.For QIP, we utilize probabilistic potential theory for the corresponding jump processes, and two essential ingredients of our proof are the tightness estimate and the Hölder regularity of caloric functions for non-elliptic stable-like processes on graphs.On the other hand, the proof of quantitative homogenization result is based on energy estimates of the localized corrector and multi-scale Poincarein equalities for non-local Dirichlet forms.

Speaker

• ### Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

AbstractWe study the Ising model on Z^3 with an externel field {\epsilon h_v} where h_v~N(0,1). We want to show that for any T lower than the critical temperature T_c, the long range order exsits as long as \epsilon is sufficiently small depending on T. This work extends previous results of Imbrie (1985) and Bricmont–Kupiainen (1988) from the very low temperature regime to the entire low tempe...

• ### Random Walks and Homogenization Theory

Description: The central-limit type results are universal in many random walk models: they are known as Donsker’s theorem for the classical Zd random walk, and also hold for some random walks in random environment, even when the environment is degenerate like percolation. In their proofs, the homogenization theory, which comes from PDE, plays an important role. This course will cover the follo...