Moderate Deviation and Exit Time Estimates for Stationary Last Passage Percolation
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Moderate Deviation and Exit Time Estimates for Stationary Last Passage Percolation Manan Bhatia1 Received: 11 May 2020 / Accepted: 2 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider planar stationary exponential last passage percolation in the positive quadrant with boundary weights. For ρ ∈ (0, 1) and points v N = ((1 − ρ)2 N , ρ 2 N ) going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case ρ = 21 in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on moderate deviation estimates for point-to-point and point-to-line exponential last passage percolation which are obtained via random matrix estimates. Keywords Last passage percolation · Moderate deviations · Exit time · TASEP · Stationary LPP
1 Introduction The planar exponential last passage percolation (LPP) model is an important and canonical integrable model in the (1+1)-dimensional KPZ universality class. The model has been mainly studied for three initial conditions—step, flat and stationary. The limiting distribution [2,12, 19], and process [1,13,20] for the passage times has been obtained by using exact formulae for different weight distributions and initial conditions. Currently, there are two main approaches for analysing the different initial conditions of the exponential LPP model—the first relying on using the random matrix connections for the models with the step and flat initial conditions to obtain concentration for the passage time [7,8,10,11]. The second approach relies on using duality along with the Burke property [14] for the stationary initial condition [4,22,26]. For the stationary initial condition, the exit time is an important quantity which has been used to establish the correct order of the variance of the stationary passage time along the characteristic line [4], along with optimal estimates for the coalescence time of two semi-
Communicated by Abhishek Dhar.
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Manan Bhatia [email protected] Department of Mathematics, Indian Institute of Science, Bangalore, India
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infinite geodesics in exponential LPP [22,26]. Estimates on the exit time have also been used as an input in investigating the time-time correlation structure in exponential LPP [6,16]. Until very recently, only suboptimal tail estimates were available for the exit time [4]. Some estimates for the exit time have also been obtained using Fredholm determinantal formulae [16,17], and the lower bound with the optimal exponent for the exit time is known [5,25]. Though there are exact correspondences to the eigenvalues of certain random matrices for the passage time in the case of the step and flat initial conditions [7,19,21], no such correspondence is known in the case of the stationary initial con
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