2023/12/05 by Van Hao Can, Can, Van Hao, Naoki Kubota +3
Mathematics · Physics and Astronomy · #82A51 #82D30 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Primary 60K37 #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #secondary 60K35
paper · pdf · doi:10.48550/arxiv.2312.02745
openalex publication_date 2023/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on \mathbb Z. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. The main object of interest in this model is the asymptotic behavior of the first passage time \rm T(0,n), which is the time needed to activate the frog at the vertex n, assuming there is only one active frog at 0 at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, Bérard and Ramírez have pointed out a slowdown phenomenon where the probability that the first passage time \rm T(0,n) is significantly larger than its expectation decays sub-exponentially and lies between exp(-n1/2+o(1)) and exp(-n1/3+o(1)). In this article, using a novel covering process approach, we confirm that 1/2 is the correct exponent, i.e., the rate of upper large deviations is given by n1/2. Moreover, we obtain an explicit rate function that is characterized by properties of Brownian motion and is strictly concave.