Tianyi Zheng

Professor — Department of Mathematics, UC San Diego

About

I am a professor in the Department of Mathematics at the University of California, San Diego since July 2022. Before coming to UCSD in the Fall of 2016, I was a postdoc in the Department of Mathematics at Stanford University. My research is in probability and geometric / asymptotic group theory, with emphasis on random walks, isoperimetry, and related aspects of dynamics and geometry.

  • Education: Ph.D. in Mathematics, Cornell University 2013; B.S., Tsinghua University 2008.
  • Research interests: Random walks on groups, isoperimetric profiles, harmonic functions, entropy, growth of groups.
Photo of Tianyi Zheng

Here is my CV (updated October 2025).

Research & Publications

Published Articles

  1. " Liouville property for groups and conformal dimension", with N. Matte Bon, V. Nekrashevych, Invent. Math. (2025). DOI
  2. "On the spectrum of asymptotic entropies of random walks", with O. Tamuz, Groups Geom. Dyn. 19 (2025), no. 3, pp. 879–897.
  3. "Prime orbit theorems for expanding Thurston maps: Dirichlet series and orbifolds", with Z. Li, Adv. Math. 443 (2024), Paper No. 109600.
  4. "Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators", with Z. Li, Adv. Math. 449 (2024), Paper No. 109723.
  5. "Prime orbit theorems for expanding Thurston maps: genericity of strong non-integrability condition", with Z. Li, Adv. Math. 450 (2024), Paper No. 109765.
  6. "Limit theorems for some long range random walks on torsion free nilpotent groups", with Z-Q. Chen, T. Kumagai, L. Saloff-Coste and J. Wang, SpringerBriefs in Mathematics, 2023.
  7. "Long range random walks and associated geometries on groups of polynomial growth." With Z-Q. Chen, T. Kumagai, L. Saloff-Coste and J. Wang, Annales de l’Institut Fourier, Volume 72 (2022) no. 3, pp. 1249-1304.
  8. "Law of large numbers for the drift of two-dimensional wreath product". with Anna Erschler, Probab. Theory Related Fields, 182, 999–1033 (2022).
  9. "Isoperimetric profiles and random walks on some groups defined by piecewise actions." With L. Saloff-Coste, Probab. Theory Relat. Fields 181, 711–756 (2021).
  10. "Speed of random walks, isoperimetry and compression of finitely generated groups." With J. Brieussel, Ann. of Math. (2) 193 (2021), no. 1, 1–105.
  11. "Random walks on discrete affine groups." With J. Brieussel and R. Tanaka, Groups, Geometry, and Dynamics 15 (3), 935-‌963, (2021).
  12. "Growth of periodic Grigorchuk groups." With A. Erschler, Invent. Math. 219 (2020), no.3, 1069 - 1155.
  13. "Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property HFD." With A. Erschler, The Annales de l’Institut Fourier (Grenoble) 70 (2020), no. 4, 1363-1402.
  14. "Random walks among time increasing conductances: heat kernel estimates." With A. Dembo and R. Huang, Probab. Theory Relat. Fields. 175 (2019), no. 1-2. 397-445.
  15. "Shalom’s property HFD and extensions by Z of locally finite groups." With J. Brieussel, Israel J. Math. 230 (2019), no. 1, 45–70.
  16. "Occupation measure of random walks and wired spanning forests in balls of Cayley graphs" With R. Lyons, Y. Peres and X. Sun. Ann. Fac. Sci. Toulouse Math. (6) 29 (2020), no. 1, 97–109.
  17. "On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds." With Y. Peres, International Mathematics Research Notices, Volume 2020, Issue 3, 722-750, February 2020.
  18. "Infinitely supported Liouville measures of Schreier graphs." With K. Juschenko, Groups Geom. Dyn. 12 (2018), no. 3, 911–918.
  19. "Isoperimetric profiles and random walks on some permutation wreath products." With L. Saloff-Coste, Rev. Mat. Iberoam. 34 (2018), no. 2, 481–540.
  20. "Random walks and isoperimetric profiles under moment conditions." With L. Saloff-Coste, Annals of Probability, 44 (2016), no. 6, 4133–4183.
  21. "Random walks on free solvable groups." With L. Saloff-Coste, Mathematische Zeitschrift, 279. 3-4 (2015): 811-848.
  22. "Random walks on nilpotent groups driven by measures supported on powers of generators." With L. Saloff-Coste, Groups, Geometry, and Dynamics 9.4 (2015): 1047-1129.
  23. "Random walks under slowly varying moment conditions on groups of polynomial volume growth", with L. Saloff-Coste. Annales de la Faculté des Sciences de Toulouse. (6) 24 (2015), no. 4, 837-855.
  24. "Large deviations for stable like random walks on Zd with applications to random walks on wreath products", with Laurent Saloff-Coste. Electronic Journal of Probability, Vol.18, no.93.
  25. "On some random walks driven by spread-out measures", with Laurent Saloff-Coste. " Groups, graphs and random walks, 444–474, London Math. Soc. Lecture Note Ser., 436, Cambridge Univ. Press, Cambridge, 2017.

Preprints

  1. "Infinite stationary measures of co-compact group actions", with M. Alhalimi, T. Hutchcroft, M. Pan, and O. Tamuz, arXiv:2410.23600 (2024), accepted for publication with the Bulletin of the London Mathematical Society.
  2. "Furstenberg entropy spectra of stationary actions of semisimple Lie groups", with J. Brieussel, arXiv preprint, arXiv:2307.01495 (2023).
  3. "Growth of groups with linear Schreier graphs", with L. Bartholdi, V. Nekrashevych, arXiv preprint, arXiv:2205.01792 (2022).
  4. "On FC-central extensions of groups of intermediate growth". arXiv preprint, arXiv:2001.07814 (2020).
  5. "Neretin groups admit no non-trivial invariant random subgroups." arXiv preprint, arXiv:1905.07605 (2019).
  6. "On rigid stabilizers and invariant random subgroups of groups of homeomorphisms." arXiv preprint, arXiv:1901.04428 (2019).

Surveys & Lecture Notes

  1. Survey "Asymptotic behaviors of random walks on countable groups", Proc. Int. Cong. Math. 2022, Vol. 4, pp. 3340–3365. EMS Press, Berlin, 2023. Survey.
  2. Minicourse notes "Random walks on Grigorchuk groups and growth estimates", IHP program Groups acting on fractals, hyperbolicity and self-similarity, SMF Panoramas and Syntheses, No. 63, 2025. Notes.

Contact

Email:

  • tzheng2 AT math.ucsd.edu

    • Department:
    Department of Mathematics
    University of California, San Diego
    9500 Gilman Drive, La Jolla, CA 92093