My research focuses on Integrable Probability, combinatorics and stochastic analysis.
Here you can find my CV.
Below you can find the list of my papers.
- Relationships between two linearizations of the box-ball system : Kerov-Kirillov-Reschetikhin bijection and slot configuration, with M. Sasada, T.Sasamoto and H.Suda
Abstract: The box-ball system (BBS), which was introduced by Takahashi and Satsuma in 1990, is a soliton cellular automaton. Its dynamics can be linearized by a few methods, among which the best known is the Kerov-Kirillov-Reschetikhin (KKR) bijection using rigged partitions. Recently a new linearization method in terms of “slot configurations” was introduced by Ferrari-Nguyen-Rolla-Wang, but its relations to existing ones have not been clarified. In this paper we investigate this issue and clarify the relation between the two linearizations. For this we introduce a novel way of describing the BBS dynamics using a carrier with seat numbers. We show that the seat number configuration also linearizes the BBS and reveals explicit relations between the KKR bijection and the slot configuration. In addition, by using these explicit relations, we also show that even in case of finite carrier capacity the BBS can be linearized via the slot configuration.
ArXiv preprint - Solvable models in the KPZ class: approach through Periodic and Free Boundary Schur measures, with T.Imamura and T.Sasamoto
Abstract: We explore probabilistic consequences of correspondences between q-Whittaker measures and periodic and free boundary Schur measures established by the authors in the recent paper [arXiv:2106.11922]. The result is a comprehensive theory of solvability of stochastic models in the KPZ class where exact formulas descend from mapping to explicit determinantal and pfaffian point processes. We discover new variants of known results as determinantal formulas for the current distribution of the ASEP on the line and new results such as Fredholm pfaffian formulas for the distribution of the point-to-point partition function of the Log Gamma polymer model in half space. In the latter case, scaling limits and asymptotic analysis allow to establish Baik-Rains phase transition for height function of the KPZ equation on the half line at the origin.
ArXiv preprint - Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials, with T.Imamura and T.Sasamoto
Abstract: Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late ’80s, we define a deterministic dynamics on the space of pairs of skew Young tableaux (P,Q). We find that this skew RSK dynamics displays conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of 0-th Kashiwara operators reveals that the skew RSK dynamics possesses symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to its linearization. Studying asymptotic evolution of the dynamics started from a pair of skew tableaux (P,Q), we discover a new bijection Υ:(P,Q)↦(V,W;κ,ν). Here (V,W) is a pair of vertically strict tableaux, i.e., column strict fillings of Young diagrams with no condition on rows, with shape prescribed by the Greene invariant, κ is an array of non-negative weights and ν is a partition.
An application of this construction is the first bijective proof of Cauchy and Littlewood identities involving q-Whittaker polynomials. New identities relating sums of q-Whittaker and Schur polynomials are also presented.
ArXiv preprint - Identity between restricted Cauchy sums for the q-Whittaker and skew Schur polynomials, with T.Imamura and T.Sasamoto
Abstract: The Cauchy identities play an important role in the theory of symmetric functions. It is known that Cauchy sums for the q-Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a q-Pochhammer symbol. We consider the sums with restrictions on the length of the first rows for labels of both polynomials and prove an identity which relates them. The proof is based on techniques from integrable probability: we rewrite the identity in terms of two probability measures: the q-Whittaker measure and the periodic Schur measure. The relation follows by comparing their Fredholm determinant formulas.
ArXiv preprint - Spin q-Whittaker polynomials and deformed quantum Toda, with L.Petrov. Communactions in Mathematical Physics
Abstract: Spin q-Whittaker symmetric polynomials labeled by partitions λ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable sl_2 vertex models. They are a one-parameter deformation of the t=0 Macdonald polynomials. We present a new more con- venient modification of spin q-Whittaker polynomials and find two Macdonald type q-difference operators acting diagonally in these polynomials with eigenvalues, respectively, q^(-λ_1) and q^(λ_N) (where λ is the polynomial’s label). We study probability measures on interlacing arrays based on spin q-Whittaker polynomials, and match their observables with known stochastic particle systems such as the q-Hahn TASEP.
In a scaling limit as q –> 1, spin q-Whittaker polynomials turn into a new one-parameter deformation of the gln Whittaker functions. The rescaled Pieri type rule gives rise to a one- parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as q –> 1 we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (arXiv:1503.04117), and relate it to spin Whittaker functions.
ArXiv preprint - Yang-Baxter random field and stochastic vertex models, with A.Bufetov and L.Petrov. Advances in Mathematics
Abstract: Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin q-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the q-Hahn PushTASEP introduced recently by Corwin-Matveev-Petrov (arXiv:1811.06475). Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the q-Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin q-Whittaker or (stable) spin Hall-Littlewood symmetric functions.
ArXiv preprint - Stationary Higher Spin Six Vertex Model and q-Whittaker measure, with T.Imamura and T.Sasamoto. Probability Theory and Related Fields.
Abstract: In this paper we consider the Higher Spin Six Vertex Model on the lattice Z≥2×Z≥1. We first identify a family of translation invariant measures and subsequently we study the one point distribution of the height function for the model with certain random boundary conditions. Exact formulas we obtain prove to be useful in order to establish the asymptotic of the height distribution in the long space-time limit for the stationary Higher Spin Six Vertex Model. In particular, along the characteristic line we recover Baik-Rains fluctuations with size of characteristic exponent 1/3. We also consider some of the main degenerations of the Higher Spin Six Vertex Model and we adapt our analysis to the relevant cases of the q-Hahn particle process and of the Exponential Jump Model.
ArXiv preprint