## Electronic Journal of Probability

### Frogs on trees?

Jonathan Hermon

#### Abstract

We study a system of simple random walks on $\mathcal{T} _{d,n}=({\cal V}_{d,n},{\cal E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda$) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o}$. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o}$. Active particles perform independent simple random walk on the tree of length $t \in{\mathbb N} \cup \{\infty \}$, referred to as the particles’ lifetime. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\mathcal{R} _t$ be the set of vertices which are visited by the process (with lifetime $t$). The susceptibility ${\mathcal S}({\mathcal T}_{d,n}):=\inf \{t:\mathcal{R} _t={\cal V}_{d,n} \}$ is the minimal lifetime required for the process to visit all sites. The cover time $\mathrm{CT} ({\mathcal T}_{d,n})$ is the first time by which every vertex was visited at least once, when we take $t=\infty$. We show that there exist absolute constants $c,C>0$ such that for all $d \ge 2$ and all $\lambda = {\lambda }_n >0$ which does not diverge nor vanish too rapidly as a function of $n$, with high probability $c \le \lambda{\mathcal S} ({\mathcal T}_{d,n}) /[n\log (n / {\lambda } )] \le C$ and $\mathrm{CT} ({\mathcal T}_{d,n})\le 3^{4\sqrt{ \log |{\cal V}_{d,n}| } }$.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 17, 40 pp.

Dates
Received: 2 October 2016
Accepted: 25 January 2018
First available in Project Euclid: 23 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1519354946

Digital Object Identifier
doi:10.1214/18-EJP144

Mathematical Reviews number (MathSciNet)
MR3771754

Zentralblatt MATH identifier
1390.60351

#### Citation

Hermon, Jonathan. Frogs on trees?. Electron. J. Probab. 23 (2018), paper no. 17, 40 pp. doi:10.1214/18-EJP144. https://projecteuclid.org/euclid.ejp/1519354946

#### References

• [1] David Aldous, Random walk covering of some special trees, J. Math. Anal. Appl. 157 (1991), no. 1, 271–283.
• [2] David Aldous, Threshold limits for cover times, J. Theoret. Probab. 4 (1991), no. 1, 197–211.
• [3] David Aldous and Jim Fill, Reversible Markov chains and random walks on graphs, 2002. Unfinished manuscript.
• [4] O. S. M. Alves, F. P. Machado, and S. Yu. Popov, Phase transition for the frog model, Electron. J. Probab. 7 (2002), no. 16, 21.
• [5] O. S. M. Alves, F. P. Machado, and S. Yu. Popov, The shape theorem for the frog model, Ann. Appl. Probab. 12 (2002), no. 2, 533–546.
• [6] O. S. M. Alves, F. P. Machado, S. Yu. Popov, and K. Ravishankar, The shape theorem for the frog model with random initial configuration, Markov Process. Related Fields 7 (2001), no. 4, 525–539.
• [7] Riddhipratim Basu, Jonathan Hermon, and Yuval Peres, Characterization of cutoff for reversible Markov chains, Ann. Probab. 45 (2017), no. 3, 1448–1487.
• [8] Itai Benjamini, Private communication, 2012.
• [9] Itai Benjamini, Luiz Renato Fontes, Jonathan Hermon, and Fabio Prates Machado, On an epidemic model on finite graphs, arXiv:1610.04301 (2016).
• [10] Itai Benjamini and Jonathan Hermon, Rapid social connectivity, arXiv:1608.07621 (2016).
• [11] Lucas Boczkowski, Yuval Peres, and Perla Sousi, Sensitivity of mixing times in Eulerian digraphs, arXiv:1603.05639 (2016).
• [12] Christian Döbler, Nina Gantert, Thomas Höfelsauer, Serguei Popov, and Felizitas Weidner, Recurrence and transience of frogs with drift on $\Bbb Z^d$, arXiv:1709.00038 (2017).
• [13] Christian Döbler and Lorenz Pfeifroth, Recurrence for the frog model with drift on $\Bbb Z^d$, Electron. Commun. Probab. 19 (2014), no. 79, 13.
• [14] Uriel Feige, A tight lower bound on the cover time for random walks on graphs, Random Structures & Algorithms 6 (1995), no. 4, 433–438.
• [15] James Allen Fill, The passage time distribution for a birth-and-death chain: strong stationary duality gives a first stochastic proof, J. Theoret. Probab. 22 (2009), no. 3, 543–557.
• [16] N. Gantert and P. Schmidt, Recurrence for the frog model with drift on $\Bbb Z$, Markov Process. Related Fields 15 (2009), no. 1, 51–58.
• [17] Arka Ghosh, Steven Noren, and Alexander Roitershtein, On the range of the transient frog model on $\Bbb{Z}$, Adv. in Appl. Probab. 49 (2017), no. 2, 327–343.
• [18] Sharad Goel, Ravi Montenegro, and Prasad Tetali, Mixing time bounds via the spectral profile, Electron. J. Probab. 11 (2006), no. 1, 1–26.
• [19] Jonathan Hermon, Ben Morris, Chuan Qin, and Allan Sly, The social network model on infinite graphs, arXiv:1610.04293 (2016).
• [20] Christopher Hoffman, Tobias Johnson, and Matthew Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635.
• [21] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Infection spread for the frog model on trees, arXiv:1710.05884 (2017).
• [22] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Recurrence and transience for the frog model on trees, Ann. Probab. 45 (2017), no. 5, 2826–2854.
• [23] Tobias Johnson and Matthew Junge, The critical density for the frog model is the degree of the tree, Electron. Commun. Probab. 21 (2016), Paper No. 82, 12.
• [24] Tobias Johnson and Matthew Junge, Stochastic orders and the frog model, arXiv:1602.04411 (2016).
• [25] O. Kallenberg, Foundations of modern probability, springer, 2002.
• [26] Samuel Karlin and James McGregor, Coincidence properties of birth and death processes, Pacific J. Math. 9 (1959), 1109–1140.
• [27] Julian Keilson, Markov chain models, rarity and exponentiality, vol. 28, Springer Science & Business Media, 2012.
• [28] Harry Kesten and Vladas Sidoravicius, The spread of a rumor or infection in a moving population, Ann. Probab. 33 (2005), no. 6, 2402–2462.
• [29] Harry Kesten and Vladas Sidoravicius, A phase transition in a model for the spread of an infection, Illinois J. Math. 50 (2006), no. 1-4, 547–634.
• [30] Harry Kesten and Vladas Sidoravicius, A shape theorem for the spread of an infection, Ann. of Math. 167 (2008), no. 3, 701–766.
• [31] Elena Kosygina and Martin P. W. Zerner, A zero-one law for recurrence and transience of frog processes, Probab. Theory Related Fields 168 (2017), no. 1-2, 317–346.
• [32] David Asher Levin and Yuval Peres, Markov chains and mixing times, American Mathematical Soc. (2017). With contributions by Elizabeth L. Wilmer and a chapter by James G. Propp and David B. Wilson. MR3726904
• [33] Eyal Lubetzky and Yuval Peres, Cutoff on all Ramanujan graphs, Geom. Funct. Anal. 26 (2016), no. 4, 1190–1216.
• [34] Laurent Miclo, On absorption times and Dirichlet eigenvalues, ESAIM Probab. Stat. 14 (2010), 117–150.
• [35] S. Yu. Popov, Frogs in random environment, J. Statist. Phys. 102 (2001), no. 1-2, 191–201.
• [36] A. F. Ramírez and V. Sidoravicius, Asymptotic behavior of a stochastic combustion growth process, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 293–334.
• [37] Joshua Rosenberg, The frog model with drift on $\Bbb R$, Electron. Commun. Probab. 22 (2017), Paper No. 30, 14.
• [38] András Telcs and Nicholas C. Wormald, Branching and tree indexed random walks on fractals, J. Appl. Probab. 36 (1999), no. 4, 999–1011.