We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles ρ ∈ (0, 1). We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.
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.is obtained as a limit of some specific two-sided random initial condition 2 . Further results for random but not necessarily stationary initial conditions are also known [29,31,38,68].For further details and recent developments around the KPZ universality class, see also the following surveys and lecture notes: [25,30,34,41,67,69,76,79].In this paper we consider a stationary model in half-space, where the latter means having a height function h(x, t) defined only on x ∈ N (or x ∈ R + ). Our model, called stationary half-space last passage percolation (LPP), is defined in Section 2.1. In this geometry there are considerably fewer results compared to the case of full-space geometry. Of course, one has to prescribe the dynamics at site x = 0. If the influence on the height function of the growth mechanism at x = 0 is very strong, then close to the origin one will essentially see fluctuations induced by it, and since the dynamics in KPZ models has to be local (in space but also in time), one will observe Gaussian fluctuations. If the influence of the origin is small, then it will not be seen in the asymptotic behavior. Between the two situations there is typically a critical value where a third different distribution function is observed. Furthermore, under a critical scaling, one obtains a family of distributions interpolating between the two extremes. For some versions of half-space LPP and related stochastic growth models (with non-random initial conditions) this has indeed been proven: one has a transition of the one-point distribution from Gaussian to GOE Tracy-Widom at the critical value, and GSE Tracy-Widom distribution [9,13,57,72] 3 Furthermore, the limit process under critical scaling around the origin is also analyzed and the transition processes have been characterized [3,4,16,72].However, the limiting distribution of the stationary LPP in half-space remained unresolved. In this paper we close this gap: in Theorem 2.3 we determine the distribution function of the stationary LPP for the finite size system and in Theorem 2.6 we determine the large time limiting distribution under critical scaling. Finally, we show that in a special limit one recovers the ...
We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence of the covariance with droplet, flat and stationary initial profile. In particular, this provides a rigorous proof of the exact formula of the covariance for the stationary case obtained in [29]. Furthermore, we prove the universality of the first order correction when the two observation times are close and provide a rigorous bound of the error term. This result holds also for random initial profiles which are not necessarily stationary.
We study the multipoint distribution of stationary half-space last passage percolation with exponentially weighted times. We derive both finite-size and asymptotic results for this distribution. In the latter case we observe a new one-parameter process we call half-space Airy stat. It is a one-parameter generalization of the Airy stat process of Baik-Ferrari-Péché, which is recovered far away from the diagonal. All these results extend the one-point results previously proven by the authors.
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