For first passage percolation on @math with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance @math , is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as @math . However the question of existence of the rate function i.e., whether the log-probability normalized by @math tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the rate function for upper tail large deviation indeed exists. Our proof can be generalized to work in higher dimensions and for the corresponding problem in last passage percolation as well. The key intuition behind the proof is that a limiting metric structure which is atypical causes the upper tail large deviation event. The formal argument then relies on an approximate version of the above which allows us to dilate the large deviation environment to compare the upper tail probabilities for various values of @math
For the upper tail large deviations, Kesten showed that The existence of the limit was left open and this open question was re-iterated in @cite_10 (See Question 18), which we answer in our Theorem .
@cite_10
2592044494 2082053835 2102318662 2298099604
We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an (o( n) ) bound by Chatterjee (Chaos, concentration, and multiple valleys, 2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (Commun. Pure Appl. Math, 2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant. We prove an upper large deviation bound for the block spin magnetization in the 2D Ising model in the phase coexistence region. The precise rate (given by the Wulff construction) is shown to hold true for all β > βc. Combined with the lower bounds derived in [I] those results yield an exact second order large deviation theory up to the critical temperature. We study the maximum of a Gaussian field on @math ( @math ) whose correlations decay logarithmically with the distance. Kahane Kah85 introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas DRSV12a DRSV12b extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in DRSV12a several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in DRSV12a : we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on the diameters of convex polyhedra. Todd and Sukegawa-Kitahara proved tail-quasipolynomial bounds on the diameters of polyhedra. These tail bounds apply when the number of facets is greater than a certain function of the dimension. We prove tail-quasipolynomial bounds on the diameters of polytopes and normal simplicial complexes. We also prove tail-polynomial upper bounds on the diameters of polyhedra.