Published October 4, 2026 | Version v2.0.0
Preprint Open

The frog model is recurrent on the 3-ary tree and transient on the 4-ary tree

Authors/Creators

Description

This is AI-generated research: the results, proofs and code were found and written by AI. Credit goes to all the humans whose work it builds on.

We prove that the frog model on the rooted $d$-ary tree is recurrent for $d=3$ and transient for $d=4$, as conjectured by Hoffman, Johnson and Junge. One frog starts awake at the root and one sleeps at every other vertex; awake frogs perform independent simple random walks, waking the frogs they visit. For $d=4$ the mean number of visits to the root is at most $5.756$. The proof runs an exact recursion on the laws of the responses of subtrees, with a law given by a finite table that dominates its image, certified by two linear systems on $159786$ states. For $d=3$ a subtree of height $m\ge1778250$ entered by one frog sends back at least $(m+2)/5$ frogs on average; the proof combines a lower model, a certified chain of steps on finite tables of bounds, and an induction on the height. Both proofs are formalized in Lean 4.

These are the cases $d=3$ and $d=4$ of Conjecture 2 of Hoffman, Johnson and Junge (Ann. Probab. 2017; arXiv:1404.6238).

The recurrence for $d=3$, the transience for $d=4$ and the bound on the expected number of visits are proved in Lean 4 with Mathlib, with no hypothesis and only the standard axioms, and Comparator with nanoda checks the three theorems against their statements; the Lean kernel checks by evaluation the certificate for $d=4$ and the run, the certificate and the table for $d=3$.

Code, Lean proofs and certificates: github.com/mt0-svg/frog-model-tree

Files

frog-model-tree.pdf

Files (125.2 MB)

Name Size Download all
md5:2bf1113dc86e5f1030804a5d0e96c803
112.6 MB Download
md5:6bb4abe03dc08fd6f71554bee61b02d6
9.9 MB Download
md5:76b0a1a436b85677bd600359a04c0f60
193.6 kB Download
md5:25a8905b82aca1bcf356319af7b101c5
1.6 MB Preview Download
md5:20fd083f0732238481ed2ddbc9cec175
732.2 kB Preview Download
md5:ace746bb3903b75ba817b277b704c56d
111.2 kB Download

Additional details

Related works

Is supplemented by
https://github.com/mt0-svg/frog-model-tree (URL)
References
10.1214/16-AOP1125 (DOI)
arXiv:1404.6238 (arXiv)

References

  • 311859884+mt0-svg@users.noreply.github.com. The frog model is recurrent on the 3-ary tree and transient on the 4-ary tree. Lean 4 formalization, Lean 4.34.1 and Mathlib v4.34.1, https://github.com/mt0-svg/frog-model-tree, 2026.
  • S. Ahmed and M. Junge. The frog model with death revisited. Preprint, arXiv:2510.18792, 2025.
  • H. Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev., 18(4):620-709, 1976.
  • C. Bailey. nanoda: an external type checker for Lean 4. Rust program, https://github.com/ammkrn/nanoda_lib, commit 3a2407216ee84a75f9e1aead6803d0578be06ae7, Sept. 2026.
  • E. Bailey, M. Junge, and J. Liu. Critical drift estimates for the frog model on trees. Electron. J. Probab., 29(52):1-21, 2024. arXiv:2303.15517.
  • E. Beckman, N. Frank, Y. Jiang, M. Junge, and S. Tang. The frog model on trees with drift. Electron. Commun. Probab., 24(26):1-10, 2019. arXiv:1808.03283.
  • B. Bond and L. Levine. Abelian networks I. Foundations and examples. SIAM J. Discrete Math., 30(2):856-874, 2016. arXiv:1309.3445.
  • K. Chatterjee, T. Quatmann, M. Schäffeler, M. Weininger, T. Winkler, and D. Zilken. Fixed point certificates for reachability and expected rewards in MDPs. In A. Gurfinkel and M. Heule, editors, Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2025, volume 15697 of Lecture Notes in Comput. Sci., pages 130-151, Cham, 2025. Springer. arXiv:2501.11467.
  • L. de Moura and S. Ullrich. The Lean 4 theorem prover and programming language. In A. Platzer and G. Sutcliffe, editors, Automated Deduction, CADE 28, volume 12699 of Lecture Notes in Comput. Sci., pages 625-635, Cham, 2021. Springer.
  • C. Döbler, N. Gantert, T. Höfelsauer, S. Popov, and F. Weidner. Recurrence and transience of frogs with drift on Z^d. Electron. J. Probab., 23(88):1-23, 2018. arXiv:1709.00038.
  • S. Franck. A phase transition for a spatial host-parasite model with extreme host immunities on Z^d and T_d. ALEA Lat. Am. J. Probab. Math. Stat., 23:63-82, 2026. arXiv:2502.08596.
  • S. Gallo and P. M. Rodriguez. Frog models on trees through renewal theory. J. Appl. Probab., 55(3):887-899, 2018. arXiv:1705.00111.
  • C. Guo, S. Tang, and N. Wei. On the minimal drift for recurrence in the frog model on d-ary trees. Ann. Appl. Probab., 32(4):3004-3026, 2022. arXiv:2008.09226.
  • J. Hermon. Frogs on trees? Electron. J. Probab., 23(17):1-40, 2018. arXiv:1609.08738.
  • C. Hoffman, T. Johnson, and M. Junge. From transience to recurrence with Poisson tree frogs. Ann. Appl. Probab., 26(3):1620-1635, 2016. arXiv:1501.05874.
  • C. Hoffman, T. Johnson, and M. Junge. Recurrence and transience for the frog model on trees. Ann. Probab., 45(5):2826-2854, 2017. arXiv:1404.6238.
  • C. Hoffman, T. Johnson, and M. Junge. Cover time for the frog model on trees. Forum Math. Sigma, 7:e41, 2019. arXiv:1802.03428.
  • T. Johnson and M. Junge. The critical density for the frog model is the degree of the tree. Electron. Commun. Probab., 21(82):1-12, 2016. arXiv:1607.07914.
  • T. Johnson and M. Junge. Stochastic orders and the frog model. Ann. Inst. Henri Poincaré Probab. Stat., 54(2):1013-1030, 2018. arXiv:1602.04411.
  • T. Johnson and L. T. Rolla. Sensitivity of the frog model to initial conditions. Electron. Commun. Probab., 24(29):1-9, 2019. arXiv:1809.03082.
  • T. Kamae, U. Krengel, and G. L. O'Brien. Stochastic inequalities on partially ordered spaces. Ann. Probab., 5(6):899-912, 1977.
  • E. Kosygina and M. P. W. Zerner. A zero-one law for recurrence and transience of frog processes. Probab. Theory Related Fields, 168(1-2):317-346, 2017. arXiv:1508.01953.
  • P. Mathews, Jr. Improved critical drift estimates for the frog model on trees. Electron. Commun. Probab., 29(41):1-14, 2024. arXiv:2309.14443.
  • M. Michelen and J. Rosenberg. The frog model on non-amenable trees. Electron. J. Probab., 25(49):1-16, 2020. arXiv:1910.05133.
  • M. Michelen and J. Rosenberg. The frog model on Galton-Watson trees. Ann. Appl. Probab., 34(4):3911-3942, 2024. arXiv:1910.02367.
  • S. Müller. A criterion for transience of multidimensional branching random walk in random environment. Electron. J. Probab., 13(41):1189-1202, 2008. arXiv:0705.1874.
  • S. Müller and G. M. Wiegel. On transience of frogs on Galton-Watson trees. Electron. J. Probab., 25(152):1-30, 2020. arXiv:2002.12008.
  • J. R. Norris. Markov Chains, volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1997.
  • C. A. d. C. Pena. Bounds on the critical parameter for the frog model on trees. PhD thesis, Universidade de São Paulo, 2024.
  • S. Y. Popov. Frogs and some other interacting random walks models. In Discrete Random Walks (DRW'03), volume AC of Discrete Math. Theor. Comput. Sci. Proc., pages 277-288, 2003.
  • J. Rosenberg. Recurrence of the frog model on the 3,2-alternating tree. ALEA Lat. Am. J. Probab. Math. Stat., 15:811-836, 2018. arXiv:1701.02813.
  • The Comparator contributors. Comparator: a judge for Lean proofs. https://github.com/leanprover/comparator, commit d03acab154d269c06e60e4de7e4cc85deebff94b, 2026.
  • The mathlib Community. The Lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), pages 367-381, New York, 2020. ACM.
  • The PARI Group. PARI/GP version 2.17.4. Univ. Bordeaux, http://pari.math.u-bordeaux.fr/, 2026.