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Abstract We study random domino tilings of the Aztec diamond with a biased$$2 \times 2$$ $2\times 2$periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight. 

Abstract We introduce and study a one parameter deformation of the polynuclear growth (PNG) in (1+1)-dimensions, which we call the $$t$$-PNG model. It is defined by requiring that, when two expanding islands merge, with probability $$t$$ they sprout another island on top of the merging location. At $t=0$, this becomes the standard (non-deformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the $$t$$-PNG model allows errors to occur in the sorting algorithm with probability $$t$$. We prove that the $$t$$-PNG model exhibits one-point Tracy–Widom Gaussian Unitary Ensemble asymptotics at large times for any fixed $$t\in [0,1)$$, and one-point convergence to the narrow wedge solution of the Kardar–Parisi–Zhang equation as $$t$$ tends to $$1$$. We further construct distributions for an external source that are likely to induce Baik–Ben Arous–Péché-type phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions. 

Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.