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Register a new userA variational formula for large deviations in First-passage percolation under tail estimates
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Consider first passage percolation with identical and independent weight distributions and first passage time ${rm T}$. In this paper, we study the upper tail large deviations $mathbb{P}({rm T}(0,nx)>n(mu+xi))$, for $xi>0$ and $x eq 0$ with a time constant $mu$ and a dimension $d$, for weights that satisfy a tail assumption $ beta_1exp{(-alpha t^r)}leq mathbb P(tau_e>t)leq beta_2exp{(-alpha t^r)}.$ When $rleq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp{(-(2dxi +o(1))n)}$. When $1< rleq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${rm T}(0,nx)>n(mu+xi)$ is described by a localization of high weights around the origin. The picture changes for $rgeq d$ where the configuration is not anymore localized.
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In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${rm T}$. Our main results prove that for any $xi>0$ and $x eq 0$, $mathbb{P}({rm T}(0,nx)>n(mu+xi))$ decays as $exp{(-(2dxi +o(1))n)}$ with a time constant $mu$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.
We study first-passage percolation where edges in the left and right half-planes are assigned values according to different distributions. We show that the asymptotic growth of the resulting inhomogeneous first-passage process obeys a shape theorem, and we express the limiting shape in terms of the limiting shapes for the homogeneous processes for the two weight distributions. We further show that there exist pairs of distributions for which the rate of growth in the vertical direction is strictly larger than the rate of growth of the homogeneous process with either of the two distributions, and that this corresponds to the creation of a defect along the vertical axis in the form of a `pyramid.
These lecture notes are written as reference material for the Advanced Course Hydrodynamical Methods in Last Passage Percolation Models, given at the 28th Coloquio Brasileiro de Matematica at IMPA, Rio de Janeiro, July 2011.
In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the Baik--Rains family. We derive our results using a related integrable model having Pfaffian structure together with careful analytic continuation and steepest descent analysis.
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $pi_p(n) / pi_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at mesoscopic distance from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter $n$ box on scale $n^{d-6}$; this result complements a lower bound of Aizenman.
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