A simple proof of the DPRZ theorem for 2d cover times

Abstract

The ε-cover time of the two dimensional unit torus T 2 by Brownian motion (BM) is the time for the process to come within distance ε > 0 from any point. Denoting by T ε (x) the first time BM hits the ε-ball centered in x ∈ T 2 , the ε-cover time is thus given by new) road to the DPRZ-TheoremWe identify the unit torus T 2 with [0, 1) × [0, 1) ⊂ R 2 , endowed with the metric d T 2 (x, y) = min {||x − y + (e 1 , e 2 ) || : e 1 , e 2 ∈ {−1, 0, 1}} .We construct BM on T 2 by W t ≡ Ŵ 1 (t) mod 1,Ŵ 2 (t) mod 1 , whereŴ is standard BM on R 2 .By monotonicity of T ε and Borel-Cantelli Lemma, the DPRZ-Theorem steadily follows from 1 arXiv:1805.09744v1 [math.PR]

First passage percolation in the mean field limit

“…In spite of the strong, inbuilt correlations, one can show that to leading order, the GREM(n) still behaves like its REM-counterpart, to wit Contrary to the GREM(2)-case, a proof of Theorem 3 is rather involved, and relies on a tool which has only recently crystallysed in the field of disordered systems, the multiscale refinement of the second moment method [35]. This is a flexible tool which has played a major role in the study of the extreme values of highly correlated random structures, such as the issue of cover times [10,45,11], the extreme values of the Riemann zeta function on the critical line [7,6], the maximum of the characteristic polynomial of random matrices [5,21], the Ginzburg-Landau model [12], and much more, see also [4] and references therein.…”