Two lattices with equal partition functions. This leads to a refined Cauchy identity

Refined Cauchy identity for spin Hall-Littlewood symmetric rational functions

[2020/07/20]

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $\mathsf{F}_\lambda$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two $\mathsf{F}_\lambda$’s as a determinant. The determinant is of Izergin-Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang-Baxter equation.

We rewrite our Izergin-Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall-Littlewood rational functions with (the Hall-Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin-Korepin type determinant yields a deformation of Schur symmetric polynomials.

In the spin-$\frac12$ specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar-Parisi-Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.

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An example of the Yang-Baxter equation in the Schur case

Spin deformation of q-Whittaker polynomials and Whittaker functions

Spin $q$-Whittaker symmetric polynomials labeled by partitions $\lambda$ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-\lambda_1}$ and $q^{\lambda_N}$ (where $\lambda$ 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 $q\nearrow 1$, spin $q$-Whittaker polynomials turn into a new one-parameter deformation of the $\mathfrak{gl}_n$ 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 $q\nearrow 1$ we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (arXiv:1503.04117), and relate it to spin Whittaker functions.

Based on a joint work with Matteo Mucciconi and earlier works with Alexey Bufetov and both of them.

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The new deformed quantum Toda Hamiltonian

Spin q-Whittaker polynomials and deformed quantum Toda

[2020/03/31]

Spin $q$-Whittaker symmetric polynomials labeled by partitions $\lambda$ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) in the context of integrable $\mathfrak{sl}_2$ vertex models. They are a one-parameter deformation of the $t=0$ Macdonald polynomials. We present a new, more convenient modification of spin $q$-Whittaker polynomials and find two Macdonald type $q$-difference operators acting diagonally in these polynomials with eigenvalues, respectively, $q^{-\lambda_1}$ and $q^{\lambda_N}$ (where $\lambda$ 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 $q\nearrow 1$, spin $q$-Whittaker polynomials turn into a new one-parameter deformation of the $\mathfrak{gl}_n$ 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 $q\nearrow 1$ we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (arXiv:1503.04117), and relate it to spin Whittaker functions.

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Chebyshev polynomials

MATH 7310 • Real Analysis and Linear Spaces I

[Spring 2020 semester]
Real part of a particularly complex function

MATH 3340 • Complex Variables with Applications

[Spring 2020 semester]
Perturbed Dyson Brownian motion - eigenvalue dynamics of a 6x6 matrix of Brownian motions, with no drift off the diagonal, and an arithmetic progression of drifts on the diagonal

Parameter symmetry in perturbed GUE corners process and reflected drifted Brownian motions

[2019/12/17]

The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix $G+\mathrm{diag}(\mathbf{a})$, where $G$ is the random matrix from the Gaussian Unitary Ensemble (GUE), and $\mathrm{diag}(\mathbf{a})$ is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression.

The construction we present may be viewed as a random matrix analogue of the recent results of the first author and Axel Saenz.

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