We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall.We observe frozen and liquid regions, prove convergence of the local correlations to translation-invariant Gibbs measures in the liquid region, and obtain new discrete Jacobi and symmetric Pearcey determinantal point processes near the wall.The model can be viewed as the one-parameter family of Plancherel measures for the infinite-dimensional orthogonal group, and we use this interpretation to derive the determinantal formula for the correlation functions at any finite time moment.

We study a two-dimensional family of probability measures on infinite Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters of the infinite-dimensional unitary group. These measures are unitary group analogs of the well-known Plancherel measures for symmetric groups.We show that any measure from our family defines a determinantal point process on Z+ × Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes.

This paper introduces a multi-species version of a process called ASEP (q, j). In this process, up to 2j particles are allowed to occupy a lattice site, the particles drift to the right with asymmetry q 2j ∈ (0.1), and there are n − 1 species of particles in which heavier particles can force lighter particles to switch places. Assuming closed boundary conditions, we explicitly write the reversible measures and a self-duality function, generalizing previously known results for two-species ASEP and single-species ASEP(q, j).Additionally, it is shown that this multi-species ASEP(q, j) is dual to its space-reversed version, in which particles drift to the left. As j → ∞, this multi-species ASEP(q, j) converges to a multi-species q-TAZRP and the self-duality function has a non-trivial limit, showing that this multi-species q-TAZRP satisfies a space-reversed self-duality.The construction of the process and the proofs are accomplished utilizing spin j representations of Uq(gl n ), extending the approach used for single-species ASEP(q, j). arXiv:1605.00691v1 [math.PR] 2 May 2016 then the duality should reduce to the ASEP(q, j) duality of [8] corresponding to the ith projection. Or, in other words, each ith class particle in ξ counts 1, . . . , ith class particles in η according to the ASEP(q, j) duality. Indeed, this paper will prove that this is indeed true.Additionally, in the j → ∞ limit, the asymmetry parameter converges to 0 and an arbitrary number of particles are allowed at each site. Indeed, the ASEP(q, j) process converges to the q-TAZRP (Totally Asymmetric Zero Range Process) introduced in [26]. A similar statement holds here: namely, the multispecies ASEP(q, j) process in this paper converges to a multi-species q-TAZRP constructed in [31]. Additionally, by applying a charge-parity symmetry, the self-duality function converges to a duality function between this multi-species q-TAZRP and its space-reversed version, in which particles jump in the opposite direction.During the preparation of this paper, the author learned in a private communication that an upcoming paper by V. Belitsky and G. Schütz [5] analyzes the n-species ASEP with closed boundary conditions (corresponding to j = 1/2 in the notation here) with Uq(gl n+1 ) symmetry and explicitly derives all invariant measures and a class of self-duality functions. Additionally, an application to the dynamics of shocks is proved.The reminder of this paper is organized as follows. Section 2 defines the process and states the results for duality and reversible measures. Section 3 goes over the [8] framework and the necessary representation theory background. Section 4 constructs the process using this framework. Section 5 proves the theorems concerning duality and reversible measures.Acknowledgments.

A recent paper [KMMO16] introduced the stochastic Uq(A (1) n ) vertex model. The stochastic S-matrix is related to the R-matrix of the quantum group Uq(A

We study low regularity behavior of the nonlinear wave equation in R 2 augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data (u, u t ) in H s (R 2 ) × H s−1 (R 2 ), is ill-posed whenever 0 < s < s cr , where the critical exponent s cr depends on the degree of nonlinearity. In particular, for the quintic nonlinearity u 5 , the critical exponent in R 2 is s cr = 1/2, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents s such that −1/6 < s ≤ s cr = 1/2. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.