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Coloured stochastic vertex models and their spectral theory

This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following: …
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Shift‐invariance for vertex models and polymers

We establish a symmetry in a variety of integrable stochastic systems: certain multi‐point distributions of natural observables are unchanged under a shift of a subset of observation points. The…
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A New Generalisation of Macdonald Polynomials

We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters (q, t) and polynomial in a further two parameters (u, v). We evaluate…
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Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries

We prove that the normalisation of the stationary state of the multi-species asymmetric simple exclusion process (mASEP) is a specialisation of a Koornwinder polynomial. As a corollary we obtain that…
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Colored Fermionic Vertex Models and Symmetric Functions

In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra Uq (…
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Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons

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Between the stochastic six vertex model and Hall-Littlewood processes

We prove that the joint distribution of the values of the height function for the stochastic six vertex model in a quadrant along a down-right path coincides with that for the lengths of the first…
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Nonsymmetric Macdonald polynomials via integrable vertex models

Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the…
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Modified Macdonald Polynomials and Integrability

We derive combinatorial formulae for the modified Macdonald polynomial $$H_{\lambda }(x;q,t)$$ H λ ( x ; q , t ) using coloured paths on a square lattice with quasi-cylindrical boundary conditions.…
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Observables of coloured stochastic vertex models and their polymer limits

In the context of the coloured stochastic vertex model in a quadrant, we identify a family of observables whose averages are given by explicit contour integrals. The observables are certain linear…
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