Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Harmonic measure in the presence of a spectral gap

Itai Benjamini and Ariel Yadin

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Abstract

We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting point is not more than a constant factor of the uniform measure on the set. For large sets there is a tight logarithmic correction factor. We also show that positive spectral gap does not allow for a fixed proportion of the harmonic measure of sets to be supported on small subsets, in contrast to the situation in Euclidean space. The results are quantitative as a function of the spectral gap, and apply also when the spectral gap decays to $0$ as the size of the graph grows to infinity. As an application we consider a model of diffusion limited aggregation, or $\mathsf{DLA}$, on finite graphs, obtaining upper bounds on the growth rate of the aggregate.

Résumé

On étudie la mesure harmonique sur les graphes finis en s’intéressant de près au cas des expanseurs, c’est à dire des graphes dont le trou spectral est positif. On montrera que dans ce cas, pour tout sous-ensemble pas trop gros, la mesure harmonique vue d’un point uniforme est bornée par un facteur multiplicatif fois la mesure uniforme sur l’ensemble. Pour les gros ensembles il y a une correction logarithmique tendue. On montrera aussi que dans le cas d’un trou spectral positif, une proportion constante de la mesure harmonique ne peut pas être supportée par de petits sous-ensembles, contrairement à ce qui se passe dans le cas euclidien. Des résultats quantitatifs sont présentés en fonction de la taille du trou spectral, et s’appliquent aussi lorsque cette taille tend vers $0$ lorsque la taille du graphe tend vers l’infini. En application, on considèrera un modèle d’agrégation limitée par diffusion ($\mathsf{DLA}$) sur des graphes finis, pour obtenir des bornes supérieures sur la croissance de l’aggrégat.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1050-1060.

Dates
Received: 26 August 2014
Revised: 4 February 2015
Accepted: 8 February 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723510

Digital Object Identifier
doi:10.1214/15-AIHP670

Mathematical Reviews number (MathSciNet)
MR3531699

Zentralblatt MATH identifier
1347.05229

Subjects
Primary: 05C81: Random walks on graphs 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85]

Keywords
Harmonic measure Spectral gap Buerling estimate DLA

Citation

Benjamini, Itai; Yadin, Ariel. Harmonic measure in the presence of a spectral gap. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1050--1060. doi:10.1214/15-AIHP670. https://projecteuclid.org/euclid.aihp/1469723510


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References

  • [1] D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. Unpublished book, 1999. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [2] N. Alon and J. H. Spencer. The Probabilistic Method. Wiley, Hoboken, NJ, 2004.
  • [3] M. T. Barlow, R. Pemantle and E. A. Perkins. Diffusion limited aggregation on a tree. Probab. Theory Related Fields 107 (1997) 1–60.
  • [4] I. Benjamini. On the support of harmonic measure for the random walk. Israel J. Math. 100 (1997) 1–6.
  • [5] I. Benjamini, H. Finucane and R. Tessera. On the scaling limit of finite vertex transitive graphs with large diameter. Combinatorica. To appear. Available at arXiv:1203.5624.
  • [6] J. Bourgain. On the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math. 87 (3) (1987) 477–483.
  • [7] R. Eldan. Diffusion limited aggregation on the hyperbolic plane. Preprint, 2013. Available at arXiv:1306.3129.
  • [8] J. B. Garnett and D. E. Marshall. Harmonic Measure, 2. Cambridge Univ. Press, Cambridge, 2005.
  • [9] P. W. Jones and T. H. Wolff. Hausdorff dimension of harmonic measures in the plane. Acta Math. 161 (1) (1988) 131–144.
  • [10] H. Kesten. How long are the arms in DLA? J. Phys. A: Math. Gen. 20 (1) (1987) L29–L33.
  • [11] H. Kesten. Upper bounds for the growth rate of DLA. Phys. A 168 (1) (1990) 529–535.
  • [12] G. F. Lawler. A discrete analogue of a theorem of Makarov. Combin. Probab. Comput. 2 (2) (1993) 181–199.
  • [13] G. F. Lawler. Intersections of Random Walks. Springer, New York, 2013.
  • [14] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. AMS, Providence, RI, 2009.
  • [15] N. G. Makarov. On the distortion of boundary sets under conformal mappings. Proc. London Math. Soc. (3) 51 (2) (1985) 369–384.
  • [16] T. A. Witten and L. M. Sander. Diffusion-limited aggregation. Phys. Rev. B 27 (9) (1983) 5686–5697.
  • [17] W. Woess. Lamplighters, Diestel–Leader graphs, random walks, and harmonic functions. Combin. Probab. Comput. 14 (3) (2005) 415–433.