Coloured stochastic vertex models and their spectral theory

Abstract

This work is dedicated to sl n+1 -related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following:(1) We construct the basis of (rational) eigenfunctions of the coloured transfer-matrices as partition functions of our lattice models with certain boundary conditions. Similarly, we construct a dual basis and prove the corresponding orthogonality relations and Plancherel formulae.(2) We derive a variety of combinatorial properties of those eigenfunctions, such as branching rules, exchange relations under Hecke divided-difference operators, (skew) Cauchy identities of different types, and monomial expansions.(3) We show that our eigenfunctions are certain (non-obvious) reductions of the nested Bethe Ansatz eigenfunctions.(5) We demonstrate how the coloured-uncoloured match degenerates to the coloured (or multi-species) versions of the ASEP, q-PushTASEP, and the q-boson model.(6) We show how our eigenfunctions relate to non-symmetric Cherednik-Macdonald theory, and we make use of this connection to prove a probabilistic matching result by applying Cherednik-Dunkl operators to the corresponding non-symmetric Cauchy identity. ContentsChapter 1. Introduction 1.1. Preface 1.2. The model 1.3. The transfer-matrix and its eigenfunctions 1.4. Plancherel theory 1.5. Summation identities, recursive relations, monomial expansions 1.6. Matching distributions 1.7. Matching for interacting particle systems 1.8. Asymptotics 1.9. A word about extensions 1.10. Acknowledgments Chapter 2. Rank-n vertex models 2.1. Stochastic U q ( sl n+1 ) R-matrix 2.2. Higher-spin L and M -matrices 2.3. Intertwining equations 2.4. Colour-blindness results 2.5. Stochastic weights Chapter 3. Row operators and non-symmetric rational functions 3.1. Space of states and row operators 3.2. Commutation relations 3.3. Coloured compositions 3.4. The rational non-symmetric functions f µ and g µ 3.5. Permuted boundary conditions 3.6. Pre-fused functions Chapter 4. Branching rules and summation identities 4.1. Skew functions f µ/ν and g µ/ν 4.2. Branching rules 4.3. Summation identities of Mimachi-Noumi type 4.4. Symmetric rational function G µ/ν 4.5. Cauchy identities Chapter 5. Recursive properties and symmetries 5.1. Factorization of f δ 5.2. Hecke algebra and its polynomial representation 5.3. Exchange relations for f µ 5.4. Symmetry in (x i , x i+1) for µ i = µ i+1 5.5. Relationship between f µ and f σ δ 5.6. Relationship between f µ and g µ5.7. Relationship between f σ δ and g µ 5.8. Exchange relations for g µ 5.9. Reduction to non-symmetric Hall-Littlewood polynomials 5.10. Eigenrelation for the non-symmetric Hall-Littlewood polynomials Chapter 6. Monomial expansions: permutation graphs 6.1. Warm-up: F -matrices for two-site spin chains 6.2. N -site R-matrices 6.3. Permutation graphs 6.4. Reversed permutation graphs 6.5. F -matrices for spin chains of generic length 6.6. Column operators 6.7. Monomial expansions Chapter 7. Monomial expansions: degenerations of nested Bethe ve...

Harold Widom's work in random matrix theory

“…Over the decade or so since Tracy and Widom's initial work on ASEP, there have been a number of developments in the field of integrable probability in this direction. This is too vast a subject to try to survey here, so we simply refer to a few existing surveys [ABW,Bor2,BoGo,BoPe1,BoPe2,BoWh,Cor1,Cor2,Cor3,Cor4,OCon,QuSp,Zyg] and mention some general topics: Kardar-Parisi-Zhang universality class, replica method, Markov duality, quantum Toda Hamiltonian, Macdonald processes (and limits to Whittaker, Jack, Hall-Littlewood or Schur processes), stochastic vertex models, spin q-Whittaker and Hall-Littlewood functions. The list goes on, as does the influence of Tracy and Widom's work on ASEP and random matrices.…”

A Vertex Model for Supersymmetric LLT Polynomials

“…While the study of integrable systems is a classical subject (see [3,26] for example), they have recently enjoyed an advent into the world of (non)symmetric polynomials [10,29,6,28]. Also known as vertex models, ice models, or multiline queues, these models have been generalized to colored vertex models [5,8,7,11,13,16] and polyqueue tableaux [13,2].…”

Shuffle algebras, lattice paths and the commuting scheme

“…We conclude this section by reformulating more explicitly the exchange relations of Lemma 2 in terms of the F v . The next proposition will not be used in what follows, but is included to reconnect to the existing literature [BW18].…”