2019/08/25 by Elnur Emrah, Emrah, Elnur, Christopher Janjigian +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60K35 #60K37 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1908.09319
openalex publication_date 2019/08/25 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We study the macroscopic evolution of the growing cluster in the exactly\nsolvable corner growth model with independent exponentially distributed waiting\ntimes. The rates of the exponentials are given by an addivitely separable\nfunction of the site coordinates. When computing the growth process\n(last-passage times) at each site, the horizontal and vertical additive\ncomponents of the rates are allowed to also vary respectively with the column\nand row number of that site. This setting includes several models of interest\nfrom the literature as special cases. Our main result provides simple explicit\nvariational formulas for the a.s. first-order asymptotics of the growth process\nunder a decay condition on the rates. Formulas of similar flavor were\nconjectured in arXiv:math/0004082, which we also establish. Subject to further\nmild conditions, we prove the existence of the limit shape and describe it\nexplicitly. We observe that the boundary of the limit shape can develop flat\nsegments adjacent to the axes and spikes along the axes. Furthermore, we record\nthe formation of persistent macroscopic spikes and crevices in the cluster that\nare nonetheless not visible in the limit shape. As an application of the\nresults for the growth process, we compute the flux function and limiting\nparticle profile for the TASEP with the step initial condition and disorder in\nthe jump rates of particles and holes. Our methodology is based on\nconcentration bounds and estimating the boundary exit probabilities of the\ngeodesics in the increment-stationary version of the model, with the only input\nfrom integrable probability being the distributional invariance of the\nlast-passage times under permutations of columns and rows.\n