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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.