Some examples of dynamics for Gelfand-Tsetlin patterns

Warren, Jon; Windridge, Peter

doi:10.1214/ejp.v14-682

Cited by 48 publications

(83 citation statements)

References 28 publications

(37 reference statements)

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“…Similar Markov chains have been previously studied in [8] without the wall, and in [50] with a different ("symplectic") interaction with the wall. [39] as a part of a solution of a much more general problem).…”

Section: Random Surface Growthmentioning

confidence: 65%

“…Similar Markov chains have been previously studied in [8] without the wall, and in [49] The measures M t are the Fourier transforms of the distinguished one-parameter family of indecomposable characters of O(∞) (the indecomposable characters of O(∞) were classified in [38] as a part of a solution of a much more general problem). It is natural to call them the Plancherel measures.…”

Section: Introductionmentioning

confidence: 89%

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Random surface growth with a wall and Plancherel measures for O (∞)

Borodin

Kuan

2010

Comm Pure Appl Math

129

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We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall.We observe frozen and liquid regions, prove convergence of the local correlations to translation-invariant Gibbs measures in the liquid region, and obtain new discrete Jacobi and symmetric Pearcey determinantal point processes near the wall.The model can be viewed as the one-parameter family of Plancherel measures for the infinite-dimensional orthogonal group, and we use this interpretation to derive the determinantal formula for the correlation functions at any finite time moment.

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Section: Random Surface Growthmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 89%

Random surface growth with a wall and Plancherel measures for O (∞)

Borodin

Kuan

2010

Comm Pure Appl Math

129

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“…If we connect Y (N − 1; T ) to a field of random Young diagrams, then the desired statement would follow from the determinantal structure of the Schur process described in Section 3.2. The desired connection of the mixed TASEP with particle-dependent inhomogeneity to Schur processes is in well-known and follows from the column Robinson-Schensted-Knuth (RSK) correspondences (see [Ful97], [Sta01] for details on RSK, and, e.g., [Joh00], [O'C03a], [WW09] for probabilistic applications of RSK to TASEPs) or, alternatively, from the results of [BF14]. The precise connection reads as follows.…”

Section: Determinantal Structure Of Dgcgmentioning

confidence: 99%

Generalizations of TASEP in Discrete and Continuous Inhomogeneous Space

Knizel

Petrov

Saenz

2019

Commun. Math. Phys.

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We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first-or last-passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar-Parisi-Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy-Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions.A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last-passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results.The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.

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“…A unified approach to both the RSK type and the BF type fields in the half-continuous setting (details on half-continuous degenerations of random fields may be found in Appendices A.6 and A.9) was suggested in [BP16b]. In fully discrete setting, elements of BF type fields for Schur polynomials appeared in [WW09], [BF14].…”

Section: Schur Whittakermentioning

confidence: 99%

Yang–baxter Field for Spin Hall–littlewood Symmetric Functions

Bufetov¹,

Petrov

2019

Forum of Mathematics, Sigma

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Employing bijectivisation of summation identities, we introduce local stochastic moves based on the Yang-Baxter equation for Uq( sl2). Combining these moves leads to a new object which we call the spin Hall-Littlewood Yang-Baxter field -a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang-Baxter field with spin Hall-Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang-Baxter field leading to new dynamic versions of the stochastic six vertex model and of the Asymmetric Simple Exclusion Process. YANG-BAXTER FIELD FOR SPIN HALL-LITTLEWOOD SYMMETRIC FUNCTIONS 2 5.2. Bijective proof of the skew Cauchy identity 23 5.3. Markov projection of the forward transition onto first columns 24 6. Yang-Baxter field 25 6.1. Spin Hall-Littlewood measures and processes 25 6.2. Yang-Baxter field 26 7. A dynamic stochastic six vertex model 28 7.1. Dynamic vertex weights 28 7.2. A dynamic Yang-Baxter equation 31 Appendix A. Degenerations and limits 32 A.1. Hall-Littlewood degeneration and stochastic six vertex model 32 A.2. Schur degeneration and modified discrete time PushTASEP 34 A.3. Discrete time PushTASEP and Schur measures 34 A.4. Hall-Littlewood degeneration with rescaling 36 A.5. Schur degeneration with rescaling 36 A.6. Half-continuous dynamic stochastic six vertex model 36 A.7. Half-continuous stochastic six vertex model 38 A.8. Continuous time modified PushTASEP 38 A.9. Continuous time PushTASEP and (2 + 1)-dimensional Yang-Baxter dynamics 38 A.10. Half-continuous Hall-Littlewood degeneration with rescaling 42 A.11. Half-continuous Schur degeneration with rescaling 42 A.12. Rational limit t → 1 43 A.13. Limit to a dynamic version of ASEP 43 A.14. Finite vertical spin 44 Appendix B. Yang-Baxter equation 44 Appendix C. Probabilities of forward and backward Yang-Baxter moves 46 Appendix D. Another form of the skew Cauchy identity 49 Appendix E. Inhomogeneous modifications 50 References 513 That is, the waiting time till the jump is an independent exponential random variable with mean equal to (rate) −1 .

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Some examples of dynamics for Gelfand-Tsetlin patterns

Cited by 48 publications

References 28 publications

Random surface growth with a wall and Plancherel measures for O (∞)

Random surface growth with a wall and Plancherel measures for O (∞)

Generalizations of TASEP in Discrete and Continuous Inhomogeneous Space

Yang–baxter Field for Spin Hall–littlewood Symmetric Functions

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