The Annals of Probability

Random walks and isoperimetric profiles under moment conditions

Laurent Saloff-Coste and Tianyi Zheng

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Abstract

Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $\vert \cdot\vert $. We study the behavior of the probability of return for random walks driven by symmetric measures $\mu$ that are such that $\sum\rho(\vert x\vert )\mu(x)<\infty$ for increasing regularly varying or slowly varying functions $\rho$, for instance, $s\mapsto(1+s)^{\alpha}$, $\alpha\in(0,2]$, or $s\mapsto(1+\log(1+s))^{\varepsilon}$, $\varepsilon>0$. For this purpose, we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to obtain a sharp $L^{2}$-version of Erschler’s inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.

Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 4133-4183.

Dates
Received: January 2015
Revised: October 2015
First available in Project Euclid: 14 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1479114272

Digital Object Identifier
doi:10.1214/15-AOP1070

Mathematical Reviews number (MathSciNet)
MR3572333

Zentralblatt MATH identifier
1378.60019

Subjects
Primary: 60B05: Probability measures on topological spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
Random walk isoperimetric profile wreath product moment

Citation

Saloff-Coste, Laurent; Zheng, Tianyi. Random walks and isoperimetric profiles under moment conditions. Ann. Probab. 44 (2016), no. 6, 4133--4183. doi:10.1214/15-AOP1070. https://projecteuclid.org/euclid.aop/1479114272


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References

  • [1] Bakry, D., Coulhon, T., Ledoux, M. and Saloff-Coste, L. (1995). Sobolev inequalities in disguise. Indiana Univ. Math. J. 44 1033–1074.
  • [2] Bendikov, A. and Saloff-Coste, L. (2012). Random walks on groups and discrete subordination. Math. Nachr. 285 580–605.
  • [3] Bendikov, A. and Saloff-Coste, L. (2012). Random walks driven by low moment measures. Ann. Probab. 40 2539–2588.
  • [4] Blachère, S., Haïssinsky, P. and Mathieu, P. (2008). Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36 1134–1152.
  • [5] Coulhon, T. (1996). Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141 510–539.
  • [6] Coulhon, T. (2003). Heat kernel and isoperimetry on non-compact Riemannian manifolds. In Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Contemp. Math. 338 65–99. Amer. Math. Soc., Providence, RI.
  • [7] Coulhon, T. and Grigor’yan, A. (1997). On-diagonal lower bounds for heat kernels and Markov chains. Duke Math. J. 89 133–199.
  • [8] Coulhon, T., Grigor’yan, A. and Pittet, C. (2001). A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble) 51 1763–1827.
  • [9] Coulhon, T. and Saloff-Coste, L. (1993). Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoam. 9 293–314.
  • [10] Dyubina, A. G. (1999). An example of the rate of departure to infinity for a random walk on a group. Uspekhi Mat. Nauk 54 159–160.
  • [11] Erschler, A. (2003). On drift and entropy growth for random walks on groups. Ann. Probab. 31 1193–1204.
  • [12] Erschler, A. (2003). On isoperimetric profiles of finitely generated groups. Geom. Dedicata 100 157–171.
  • [13] Erschler, A. (2006). Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks. Probab. Theory Related Fields 136 560–586.
  • [14] Erschler, A. and Karlsson, A. (2010). Homomorphisms to $\mathbb{R}$ constructed from random walks. Ann. Inst. Fourier (Grenoble) 60 2095–2113.
  • [15] Gournay, A. (2014). The Liouville property via Hilbertian compression. Preprint. Available at arXiv:1403.1195.
  • [16] Hebisch, W. and Saloff-Coste, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673–709.
  • [17] Kaĭmanovich, V. A. and Vershik, A. M. (1983). Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11 457–490.
  • [18] Kotowski, M. and Virág, B. (2015). Non-Liouville groups with return probability exponent at most 1/2. Electron. Commun. Probab. 20 no. 12, 12.
  • [19] Naor, A. and Peres, Y. (2008). Embeddings of discrete groups and the speed of random walks. Int. Math. Res. Not. IMRN. Art. ID rnn 076, 34.
  • [20] Pittet, Ch. and Saloff-Coste, L. (2000). A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on cayley graphs, with examples. Unpublished manuscript.
  • [21] Pittet, Ch. and Saloff-Coste, L. (2000). On the stability of the behavior of random walks on groups. J. Geom. Anal. 10 713–737.
  • [22] Pittet, C. and Saloff-Coste, L. (2002). On random walks on wreath products. Ann. Probab. 30 948–977.
  • [23] Saloff-Coste, L. and Zheng, T. (2012). On some random walks driven by spread-out measures. Preprint. Available at arXiv:1309.6296.
  • [24] Saloff-Coste, L. and Zheng, T. (2013). Large deviations for stable like random walks on $\mathbb{Z}^{d}$ with applications to random walks on wreath products. Electron. J. Probab. 18 no. 93, 35.
  • [25] Saloff-Coste, L. and Zheng, T. (2014). Random walks under slowly varying moment conditions on groups of polynomial volume growth. Preprint. Available at arXiv:1507.03551.
  • [26] Saloff-Coste, L. and Zheng, T. (2015). Random walks on free solvable groups. Math. Z. 279 811–848.
  • [27] Saloff-Coste, L. and Zheng, T. (2015). Random walks on nilpotent groups driven by measures supported on powers of generators. Groups Geom. Dyn. 9 1047–1129.
  • [28] Schilling, R. L., Song, R. and Vondraček, Z. (2012). Bernstein Functions: Theory and Applications, 2nd ed. De Gruyter Studies in Mathematics 37. de Gruyter, Berlin.
  • [29] Schilling, R. L. and Wang, J. (2012). Functional inequalities and subordination: Stability of Nash and Poincaré inequalities. Math. Z. 272 921–936.
  • [30] Tessera, R. (2011). Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. Comment. Math. Helv. 86 499–535.
  • [31] Tessera, R. (2013). Isoperimetric profile and random walks on locally compact solvable groups. Rev. Mat. Iberoam. 29 715–737.