2022/03/03 by Viktor Bezborodov, Bezborodov, Viktor, Luca Di Persio +3
Mathematics · Physics and Astronomy · #60K35 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2203.01592
openalex publication_date 2022/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the continuous-time frog model on ℤ. At time t = 0, there are η(x) particles at x∈ ℤ, each of which is represented by a random variable. In particular, (η(x))x ∈ ℤ is a collection of independent random variables with a common distribution μ, μ(ℤ+) = 1. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. Active particles perform a simple symmetric continuous-time random walk in ℤ (that is, a random walk with exp(1)-distributed jump times and jumps -1 and 1, each with probability 1/2), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if μ is the distribution of eY ln Y with a non-negative random variable Y satisfying 𝔼 Y < ∞, then a.s. no explosion occurs. On the other hand, if a ∈ (0,1) and μ is the distribution of eX, where ℙ \X ≥ t \ = t-a, t ≥ 1, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.