An irreversible local Markov chain that preserves the six vertex model on a torus

Borodin, Alexei; Bufetov, Alexander I.

doi:10.1214/15-aihp722

Cited by 9 publications

(10 citation statements)

References 19 publications

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“…Another way of introducing locality to Markov chains X • u;{vt} and X + {ut};v that works for generic u and v, respectively, could be to consider "multivariate" chains on whole interlacing arrays (similarly to, e.g., [O'C03a], [O'C03b], [BF14], [BP13], [MP15], with [BB15] providing an application to the six vertex model on the torus), but we will not discuss this here.…”

Section: For Anymentioning

confidence: 99%

Higher spin six vertex model and symmetric rational functions

Borodin

Petrov

2016

Sel. Math. New Ser.

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124

297

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Abstract. We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes.Our arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the inhomogeneous higher spin six vertex model for suitable domains. In the homogeneous case, such functions were previously studied in [Bor14]; they also generalize classical Hall-Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang-Baxter equation for the higher spin six vertex model.

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Section: For Anymentioning

confidence: 99%

Higher spin six vertex model and symmetric rational functions

Borodin

Petrov

2016

Sel. Math. New Ser.

Self Cite

124

297

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“…Now, the third requirement of Definition 6.3 implies that (6.9) holds for all z ∈ C (1) ∪ Γ (1) . Furthermore, it quickly follows from the definition of the contours C (1) and Γ (1) (see Definition 6.3), and the fact that |α − β|, |γ − β| = O T −10 , that there exists C 5 > 0 such that 19) for all w ∈ C (1) and v ∈ Γ (1) .…”

Section: Contour Truncationmentioning

confidence: 99%

“…This will rely on certain conservation laws satisfied by the quantities N ω;Λ (i 1 , j 1 ; i 2 , j 2 ). The first four laws listed below are quickly verified (and are also explained, in the similar case of the six-vertex model on a torus, in Section 3 of [19]), so their derivations are omitted. The fifth law is a consequence of the second, third, and fourth laws.…”

Section: 2mentioning

confidence: 99%

Current fluctuations of the stationary ASEP and six-vertex model

Aggarwal¹

2018

Duke Math. J.

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Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time T , and show that they are of order T 1/3 along a characteristic line. Upon scaling by T 1/3 , we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn in 2005 (who established the same result for the TASEP), and the work of Balázs and Seppäläinen in 2010 (who established the T 1/3 scaling for the general ASEP).Second, we introduce a class of translation-invariant Gibbs measures that characterizes a oneparameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size T and show that they too are of order T 1/3 along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995 stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity.Upon scaling the height fluctuations by T 1/3 , we again recover the Baik-Rains distribution in the large T limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992. 11 3. Observables for Models With Double-Sided

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“…The latter is a consequence of certain conservation laws satisfied by the quantities N ω;Λ (i 1 , j 1 ; i 2 , j 2 ). As discussed in Appendix A of [1] we have the following conservation laws (see also Section 3 in [10]).…”

Section: 1mentioning

confidence: 99%

Six-vertex Models and the GUE-corners Process

Dimitrov

2018

International Mathematics Research Notices

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In this paper we consider a class of probability distributions on the six-vertex model from statistical mechanics, which originate from the higher spin vertex models of [18]. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. The development of our operators is largely based on the properties of a remarkable family of symmetric rational functions, which were previously studied in [9].For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a particular choice of parameters we analyze the limit of the correlation functions through a steepest descent method. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand-Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the GUE-corners process.

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An irreversible local Markov chain that preserves the six vertex model on a torus

Cited by 9 publications

References 19 publications

Higher spin six vertex model and symmetric rational functions

Higher spin six vertex model and symmetric rational functions

Current fluctuations of the stationary ASEP and six-vertex model

Six-vertex Models and the GUE-corners Process

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