2019/12/10 by Riddhipratim Basu, Shirshendu Ganguly, Basu, Riddhipratim +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Bayesian Methods and Mixture Models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.1912.04891
72 pages, 8 figures. Appendix C expanded with complete proofs of estimates on weights of paths constrained in parallelograms as well as those having high transversal fluctuations. To appear in Comm. Math. Phys
openalex publication_date 2019/12/10 · openalex created_date 2019/12/26 · arxiv created 2021/01/27 · arxiv updated 2021/01/28 · openalex updated_date 2026/07/28
We consider directed last passage percolation on ℤ2 with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn (SIGMA, 2016), we show that for the passage times from the line x+y=0 to the points (r,r) and (n,n), denoted Xr and Xn respectively, as n→ ∞ and (r)/(n) is small but bounded away from zero, the covariance satisfies Cov(Xr,Xn)=Θ(((r)/(n))4/3+o(1) n2/3), thereby establishing (4)/(3) as the temporal covariance exponent. This differs from the corresponding exponent for the droplet initial condition recently rigorously established in Ferrari and Occelli (2018), Basu and Ganguly (2018), and requires novel arguments. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are expected to be applicable for a wider class of initial data.