Stationary Half-Space Last Passage Percolation

Abstract

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 ...

Half-Space Stationary Kardar–Parisi–Zhang Equation

“…As we mentioned above, it remains to be shown that our formula is equivalent to the Fredholm Pfaffian formula obtained in [ 62 , Theorem 2.7] (setting there and ) as expected from universality.

Fig.

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“…We expect, from the universality within the KPZ class, that our present result and theirs should match. The Pfaffian formula of [ 62 , Theorem 2.7] and our formulae ( 2.25 ), ( 2.27 ) and ( 2.28 ) look different. Although we also have a Pfaffian representation, Eq.…”

“…It is important to mention that very recently Betea et al [ 62 ] studied stationary half-space KPZ growth for a discrete model, the last-passage-percolation with exponential weights (i.e. a zero-temperature polymer).…”

“…It would be interesting to study the distribution of h ( x , t ) when x is at a distance of order from 0, and A , B are scaled close to . This would correspond to varying the parameter in [ 62 ]. However, while we can obtain some integral formula for the moments of Z ( x , t ) in the case , see ( 4.19 ) below, we do not expect that it can be rewritten as a Pfaffian formula and the asymptotic analysis would require to develop other methods.…”

Mixing times for the TASEP in the maximal current phase

“…Exponential-1-weights. A stationary TASEP on N with an open boundary was investigated [49], and more recently in [2,14]. In Section 3, we provide the above equivalences for the TASEP on a segment with open boundaries; see also [38] for similar ideas in the physics literature.…”