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A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on\mathbb{Z}^{2}with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density(1/4+x/2+x^{2}/8)e^{-x}, and so is a mixture of Gamma(1,1), Gamma(2,1), and Gamma(3,1) distributions with weights1/4,1/2, and1/4respectively. More generally, we study the local environment as seen from vertices along geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from(0,0)ton\boldsymbol{\rho}for some vector\boldsymbol{\rho}in the first quadrant, in the limit asn\to\infty, as well as semi-infinite geodesics in direction\boldsymbol{\rho}. We show almost sure convergence of the empirical distributions of the environments along these geodesics, as well as convergence of the distributions of the environment around a typical point in these geodesics, to the same limiting distribution, for which we give an explicit description.We make extensive use of a correspondence with TASEP as seen from an isolated second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for last-passage times, available from the integrable probability literature. 

Abstract In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to for some small . Janowsky and Lebowitz  posed the well‐known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of is 0. This was ultimately resolved rigorously in Basu‐Sidoravicius‐Sly which established that . Here we study the effect of the current as tends to 0 and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular we show that the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal. 

Abstract Continuing our earlier work in Nam et al. (One-step replica symmetry breaking of random regular NAE-SAT I,arXiv:2011.14270, 2020), we study the random regulark-nae-satmodel in the condensation regime. In Nam et al. (2020), the (1rsb) properties of the model were established with positive probability. In this paper, we improve the result to probability arbitrarily close to one. To do so, we introduce a new framework which is the synthesis of two approaches: the small subgraph conditioning and a variance decomposition technique using Doob martingales and discrete Fourier analysis. The main challenge is a delicate integration of the two methods to overcome the difficulty arising from applying the moment method to an unbounded state space. 

Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold λ 1 \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ \xi has an exponential tail. In this paper, we derive the first-order asymptotics of λ 1 \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ \xi is appropriately concentrated around its mean, we demonstrate that λ 1 ( ξ ) ∼ 1 / E ξ \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as E ξ → ∞ \mathbb {E}\xi \rightarrow \infty , which matches with the known asymptotics on d d -regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well.