2018/07/24 by Riddhipratim Basu, Shirshendu Ganguly, Basu, Riddhipratim +1
Mathematics · Physics and Astronomy · #Brownian motion #Combinatorics #Critical exponent #Directed percolation #Exponential function #FOS: Mathematics #FOS: Physical sciences #Geometry #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Percolation (cognitive psychology) #Physics #Probability (math.PR) #Random Matrices and Applications #Scaling #Statistics #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1807.09260
26 pages, 4 figures
openalex publication_date 2018/07/24 · arxiv created 2018/08/12 · arxiv updated 2018/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For directed last passage percolation on ℤ2 with exponential passage times on the vertices, let Tn denote the last passage time from (0,0) to (n,n). We consider asymptotic two point correlation functions of the sequence Tn. In particular we consider \rm Corr(Tn, Tr) for r≤ n where r,n→ ∞ with r≪ n or n-r ≪ n. We show that in the former case \rm Corr(Tn, Tr)=Θ(((r)/(n))1/3) whereas in the latter case 1-\rm Corr(Tn, Tr)=Θ(((n-r)/(n))2/3). The argument revolves around finer understanding of polymer geometry and is expected to go through for a larger class of integrable models of last passage percolation. As by-products of the proof, we also get a couple of other results of independent interest: Quantitative estimates for locally Brownian nature of pre-limits of Airy2 process coming from exponential LPP, and precise variance estimates for lengths of polymers constrained to be inside thin rectangles at the transversal fluctuation scale.