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

Elcio Lebensztayn; Jaime Utria

doi:10.1007/s10955-019-02294-4

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arXiv · 1901.06412

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

Elcio Lebensztayn·Jaime Utria

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Abstract

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.

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Elcio Lebensztayn, Jaime Utria. 2019-01-18. A new upper bound for the critical probability of the frog model on homogeneous trees. https://doi.org/10.1007/s10955-019-02294-4

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