Stationary stochastic Higher Spin Six Vertex Model and q-Whittaker measure

Imamura, Takashi; Mucciconi, Matteo; Sasamoto, Tomohiro

doi:10.1007/s00440-020-00966-x

Cited by 19 publications

(17 citation statements)

References 76 publications

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“…For flat limit shapes and non-random initial conditions, the limit process is known as the Airy 1 process [24,71] with the GOE Tracy-Widom as one-point distribution [78], see [9,10,63]. Finally, stationary initial conditions also lead to flat limit shapes and the Airy stat process [6], having the Baik-Rains distribution as the one-point distribution [1,7,21,40,48,[50][51][52]. The stationary model stationary case (Section 3.2); we take the latter limit using a careful analytic continuation argument (Section 3.3) and this yields the finite result.…”

Section: Introductionmentioning

confidence: 99%

Stationary Half-Space Last Passage Percolation

Betea

Ferrari

Occelli

2020

Commun. Math. Phys.

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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 ...

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Section: Introductionmentioning

confidence: 99%

Stationary Half-Space Last Passage Percolation

Betea

Ferrari

Occelli

2020

Commun. Math. Phys.

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“…It would be very interesting to find an extension of the symmetric functions formalism used in [BCFV15] in order to establish Conjecture 3.11. Another way around this obstacle which leads to observables suitable for asymptotic analysis was suggested in [IS17], [IMS19]. Overall, we believe that our conjecture can be established with the help of a good notion of analytic continuation from known Fredholm determinantal formulas.…”

Section: Contour Integral Observablesmentioning

confidence: 64%

The q-Hahn PushTASEP

Corwin

Матвеев

Petrov

2019

International Mathematics Research Notices

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We introduce the q-Hahn PushTASEP -an integrable stochastic interacting particle system which is a 3-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (Totally Asymmetric Simple Exclusion Process). The transition probabilities in the q-Hahn PushTASEP are expressed through the 4φ3 basic hypergeometric function. Under suitable limits, the q-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the q-Hahn TASEP introduced by Povolotsky [Pov13]. We establish Markov duality relations and contour integral formulas for the q-Hahn PushTASEP.In a q → 1 limit of our process we arrive at a random recursion which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e., the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner.

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“…In this section, we provide a one parameter family of stationary distribution for the unfused SHS6V model. It is worth to remark that in the recent work of [IMS19], a translation-invariant Gibbs measure was obtained (using the idea from [Agg18a]) for the space-time inhomogeneous SHS6V model on the full lattice, see Proposition 4.5 of [IMS19]. However, It is not obvious that the dynamic of SHS6V model under this Gibbs measure coincides with the one of the bi-infinite SHS6V model specified in Lemma 2.1.…”

Section: Proof Of Proposition 68 Via Self-averagingmentioning

confidence: 98%

“…However, It is not obvious that the dynamic of SHS6V model under this Gibbs measure coincides with the one of the bi-infinite SHS6V model specified in Lemma 2.1. This being the case, we choose to proceed without relying on the result from [IMS19].…”

Section: Proof Of Proposition 68 Via Self-averagingmentioning

confidence: 99%