Deformed Polynuclear Growth in (1+1) Dimensions
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Author(s)
Aggarwal, Amol
Borodin, Alexei
Wheeler, Michael
Date Issued
2022
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Citation
Aggarwal, Amol, Borodin, Alexei and Wheeler, Michael. 2022. "Deformed Polynuclear Growth in (1+1) Dimensions." International Mathematics Research Notices.
Version
Original manuscript
Abstract
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.
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNAC029
