We study the frog model on $\mathbb{Z} ^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.
References
[2] 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. 1013.60081 10.1214/aoap/1026915614 euclid.aoap/1026915614[2] 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. 1013.60081 10.1214/aoap/1026915614 euclid.aoap/1026915614
[5] L. R. Fontes, F. P. Machado, and A. Sarkar, The critical probability for the frog model is not a monotonic function of the graph, J. Appl. Probab. 41 (2004), no. 1, 292–298.[5] L. R. Fontes, F. P. Machado, and A. Sarkar, The critical probability for the frog model is not a monotonic function of the graph, J. Appl. Probab. 41 (2004), no. 1, 292–298.
[8] A. Ghosh, S. Noren, and A. Roitershtein, On the range of the transient frog model on $\mathbb{Z} $, Adv. in Appl. Probab. 49 (2017), no. 2, 327–343. MR3668379 10.1017/apr.2017.3[8] A. Ghosh, S. Noren, and A. Roitershtein, On the range of the transient frog model on $\mathbb{Z} $, Adv. in Appl. Probab. 49 (2017), no. 2, 327–343. MR3668379 10.1017/apr.2017.3
[11] C. Hoffman, T. Johnson, and M. Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635. 1345.60116 10.1214/15-AAP1127 euclid.aoap/1465905013[11] C. Hoffman, T. Johnson, and M. Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635. 1345.60116 10.1214/15-AAP1127 euclid.aoap/1465905013
[12] T. Johnson and M. Junge, The critical density for the frog model is the degree of the tree, Electron. Commun. Probab. 21 (2016), 12 pp. MR3580451 10.1214/16-ECP29[12] T. Johnson and M. Junge, The critical density for the frog model is the degree of the tree, Electron. Commun. Probab. 21 (2016), 12 pp. MR3580451 10.1214/16-ECP29
[14] E. Kosygina and M. 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.[14] E. Kosygina and M. 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.
[15] G. Lawler and V. Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.[15] G. Lawler and V. Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.
[16] É. Lebensztayn, F. P. Machado, and S. Yu. Popov, An improved upper bound for the critical probability of the frog model on homogeneous trees, J. Stat. Phys. 119 (2005), no. 1-2, 331–345.[16] É. Lebensztayn, F. P. Machado, and S. Yu. Popov, An improved upper bound for the critical probability of the frog model on homogeneous trees, J. Stat. Phys. 119 (2005), no. 1-2, 331–345.
[17] S. Yu. Popov, Frogs in random environment, J. Statist. Phys. 102 (2001), no. 1-2, 191–201. 0972.60101 10.1023/A:1026516826875[17] S. Yu. Popov, Frogs in random environment, J. Statist. Phys. 102 (2001), no. 1-2, 191–201. 0972.60101 10.1023/A:1026516826875
[19] A. Telcs and N.C. Wormald, Branching and tree indexed random walks on fractals, J. Appl. Probab. 36 (1999), no. 4, 999–1011. 0967.60059 10.1239/jap/1032374750 euclid.jap/1032374750[19] A. Telcs and N.C. Wormald, Branching and tree indexed random walks on fractals, J. Appl. Probab. 36 (1999), no. 4, 999–1011. 0967.60059 10.1239/jap/1032374750 euclid.jap/1032374750