Hall–Littlewood RSK field

Bufetov, Alexander I.; Матвеев, К. А.

doi:10.1007/s00029-018-0442-y

Cited by 31 publications

(44 citation statements)

References 33 publications

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“…• We consider a number of degenerations of the Yang-Baxter field, including new dynamic versions of the stochastic six vertex model (Section 7) and the Asymmetric Simple Exclusion Process (Appendix A.13). Our results about these dynamic models generalize those of the recent works [BBW16] and [BM17]. Let us describe our results in more detail.…”

supporting

confidence: 87%

“…Along with single-layer projections leading to DS6V, one can consider multilayer projections of the full Yang-Baxter field, as was done in [BM17, Sections 4.4 and 4.5] for the Hall-Littlewood RSK field. In particular, one can check that the two-layer projection of our Yang-Baxter field, in the Hall-Littlewood degeneration, coincides with the two-layer stochastic six vertex model of [BM17,Section 4.4]. However, the corresponding degeneration of the full Yang-Baxter field is different from the full Hall-Littlewood RSK field.…”

Section: Schur Whittakermentioning

confidence: 89%

“…In comparison, an RSK type approach would likely require very nontrivial combinatorial considerations (cf. [BM17] for the Hall-Littlewood case), further complicated by the fact that the spin Hall-Littlewood functions are not homogeneous polynomials while the usual Hall-Littlewood ones are (see Remark A.2 for more details). A BF type approach, while clearly being applicable in the spin Hall-Littlewood case, might not readily produce Markov projections.…”

Section: Schur Whittakermentioning

confidence: 99%

“…The joint distribution of the values of the height function in DS6V along down-right paths 2 can be identified with that of certain observables of a spin Hall-Littlewood process (Corollary 7.4). In the degeneration turning the spin Hall-Littlewood symmetric functions into the Hall-Littlewood ones, the DS6V model becomes the usual stochastic six vertex model of [GS92], [BCG16], and Corollary 7.4 turns into the statement established in [BM17].…”

Section: Schur Whittakermentioning

confidence: 99%

“…Recently RSK type fields associated with deformations of Schur processes (see Figure 1) were constructed for Whittaker processes [O'C12], [COSZ14], [OSZ14], q-Whittaker processes [OP13], [BP16b], [Pei13], [MP15], [Pei16], and Hall-Littlewood processes [BP15], [BBW16], [BM17]. Constructions at the Whittaker level relied on the geometric (also sometimes called "tropical") lifting of the RSK correspondence [Kir01], [NY04], while the q-Whittaker and Hall-Littlewood developments required nontrivial randomizations of the original RSK insertion algorithm.…”

mentioning

confidence: 99%

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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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supporting

confidence: 87%

Section: Schur Whittakermentioning

confidence: 89%

Section: Schur Whittakermentioning

confidence: 99%

Section: Schur Whittakermentioning

confidence: 99%

mentioning

confidence: 99%

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Yang–baxter Field for Spin Hall–littlewood Symmetric Functions

Bufetov¹,

Petrov

2019

Forum of Mathematics, Sigma

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Limits and fluctuations of p-adic random matrix products

Peski¹

2021

Sel. Math. New Ser.

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We study the distribution of singular numbers of products of certain classes of padic random matrices, as both the matrix size and number of products go to ∞ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on Z, defined explicitly in terms of certain intricate mixed q-series/exponential sums. This object may be viewed as a nontrivial p-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg [6], which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from GLN (Zp), and the p-adic analogue of Dyson Brownian motion studied in [80]. Contents 1. Introduction 1 2. p-adic matrix background 3. Symmetric function background 4. The limit of the Plancherel/principal Hall-Littlewood measure 5. Residue expansions 6. Tightness and the limiting random variable 7. An indeterminate moment problem 8. Examples of residue formula for L k,t,χ 9. The case of pure α specializations 10. From processes of infinitely many particles to finite matrix bulk limits Appendix A. Parallels with complex matrix products Appendix B. A Hall-Littlewood proof of [80, Theorem 1.2] References

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Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

Gavrilova,

Petrov

2024

Sel. Math. New Ser.

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We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the $$4\times 4$$ 4 × 4 problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size $$n\ge 4$$ n ≥ 4 , which appear new for $$n\ge 5$$ n ≥ 5 . By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.

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Hall–Littlewood RSK field

Cited by 31 publications

References 33 publications

Yang–baxter Field for Spin Hall–littlewood Symmetric Functions

Yang–baxter Field for Spin Hall–littlewood Symmetric Functions

Limits and fluctuations of p-adic random matrix products

Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

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