arXiv subjects

Jaime Utria

Publications and source records attributed to Jaime Utria.

3 recordsLinked to original sources

arXiv2026

The frog model with death and drift on free products of complete graphs

We study the frog model with death and drift on $\mathbb{D}_{m,d}$, the free product of $d+1$ copies of the complete graph of order $m$. Active and inactive particles are located at the vertices of $\mathbb{D}_{m,d}$. Each active particle performs a $\alpha$-biased random walk towards the root of $\mathbb{D}_{m,d}$, dying after a random lifetime with a geometric distribution of parameter $1-p$. Each inactive particle remains dormant until an active particle visits its location. We present conditions on the parameters $\alpha$ and $p$ for the process to die out almost surely and to survive with positive probability. Our proofs are based on comparisons of the model with simple and multi-type branching processes.

math.PR↗

arXiv2019

A new upper bound for the critical probability of the frog model on homogeneous trees

We consider the interacting particle system on the homogeneous tree of degree $(d + 1)$, known as frog model. In this model, active particles perform independent random walks, awakening all sleeping particles they encounter, and dying after a random number of jumps, with geometric distribution. We prove an upper bound for the critical parameter of survival of the model, which improves the previously known results. This upper bound was conjectured in a paper by Lebensztayn et al. ($ J. Stat. Phys.$, 119(1-2), 331-345, 2005). We also give a closed formula for the upper bound.

math.PR↗

arXiv2018

Phase transition for the frog model on biregular trees

We study the frog model with death on the biregular tree $\mathbb{T}_{d_1,d_2}$. Initially, there is a random number of awake and sleeping particles located on the vertices of the tree. Each awake particle moves as a discrete-time independent simple random walk on $\mathbb{T}_{d_1,d_2}$ and has a probability of death $(1-p)$ before each step. When an awake particle visits a vertex which has not been visited previously, the sleeping particles placed there are awakened. We prove that this model undergoes a phase transition: for values of $p$ below a critical probability $p_c$, the system dies out almost surely, and for $p > p_c$, the system survives with positive probability. We establish explicit bounds for $p_c$ in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies $p_c(\mathbb{T}_{d_1,d_2}) = 1/2 + \Theta(1/d_1+1/d_2)$ as $d_1, d_2 \to \infty$.

math.PR↗