On a class of random perturbations of the hierarchical Laplacian

A. D. Bendikov ; A. A. Grigor'yan ; S. A. Molchanov ; G. P. Samorodnitsky

Izvestiya. Mathematics , Tome 79 (2015) no. 5, pp. 859-893.

Résumé

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set $B$ of all balls of positive measure of $X$, we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts on $L^{2}(X,m)$. It is essentially self-adjoint and has a pure point spectrum. By choosing a family $\{\varepsilon (B)\}$ of independent identically distributed random variables, we define the perturbed function $C(B,\omega)$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega)}$. We study the arithmetic means $\bar{\lambda }(\omega)$ of the eigenvalues of $L^{\omega }$. Under some mild assumptions the normalized arithmetic means $( \bar{\lambda }-\mathbb{E}\bar{\lambda })/\sigma [\bar{\lambda }]$ converge to $N(0,1)$ in distribution. We also give examples when the normal convergence fails. We prove the existence of an integrated density of states. Introducing an empirical point process $N^{\omega }$ for the eigenvalues of $L^{\omega }$ and assuming that the density of states exists and is continuous, we prove that the finite-dimensional distributions of $N^{\omega }$ converge to those of the Poisson point process. As an example we consider random perturbations of the Vladimirov operator acting on $L^{2}(X,m)$, where $X=\mathbb{Q}_{p}$ is the ring of $p$-adic numbers and $m$ is the Haar measure.

Détail

Keywords: ultrametric measure space, field of $p$-adic numbers, hierarchical Laplacian, fractional derivative, Vladimirov Laplacian, point spectrum, integrated density of states, Erdős problem, point process
Mots-clés : Bernoulli convolutions, Poisson convergence.

Bibliographie

[1] F. J. Dyson, “The dynamics of a disordered linear chain”, Phys. Rev. (2), 92:6 (1953), 1331–1338 | DOI | MR | Zbl

[2] F. J. Dyson, “Existence of a phase-transition in a one-dimensional Ising ferromagnet”, Comm. Math. Phys., 12:2 (1969), 91–107 | DOI | MR | Zbl

[3] A. Bovier, “The density of states in the Anderson model at weak disorder: a renormalization group analysis of the hierarchical model”, J. Statist. Phys., 59:3-4 (1990), 745–779 | DOI | MR | Zbl

[4] E. Kritchevski, “Hierarchical Anderson model”, Probability and mathematical physics, CRM Proc. Lecture Notes, 42, Amer. Math. Soc., Providence, RI, 2007, 309–322 | MR | Zbl

[5] E. Kritchevski, “Spectral localization in the hierarchical Anderson model”, Proc. Amer. Math. Soc., 135:5 (2007), 1431–1440 | DOI | MR | Zbl

[6] E. Kritchevski, “Poisson statistics of eigenvalues in the hierarchical Anderson model”, Ann. Henri Poincaré, 9:4 (2008), 685–709 | DOI | MR | Zbl

[7] S. Molchanov, “Hierarchical random matrices and operators. Application to Anderson model”, Multidimensional statistical analysis and theory of random matrices (Bowling Green, OH, 1996), VSP, Utrecht, 1996, 179–194 | MR | Zbl

[8] S. Albeverio, W. Karwowski, “A random walk on $p$-adics – the generator and its spectrum”, Stochastic Process. Appl., 53:1 (1994), 1–22 | DOI | MR | Zbl

[9] J. Pearson, J. Bellissard, “Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets”, J. Noncommut. Geom., 3:3 (2009), 447–480 | DOI | MR | Zbl

[10] M. Aizenman, S. A. Molchanov, “Localization at large disorder and at extreme energies: an elementary derivation”, Comm. Math. Phys., 157:2 (1993), 245–278 | DOI | MR | Zbl

[11] N. Minami, “Local fluctuation of the spectrum of a multidimensional Anderson tight binding model”, Comm. Math. Phys., 177:3 (1996), 709–725 | DOI | MR | Zbl

[12] A. Bendikov, A. Grigor'yan, C. Pittet, “On a class of Markov semigroups on discrete ultra-metric spaces”, Potential Anal., 37:2 (2012), 125–169 | DOI | MR | Zbl

[13] A. Bendikov, A. Grigoryan, C. Pittet, W. Woess, “Isotropic Markov semigroups on ultra-metric spaces”, Russian Math. Surveys, 69:4 (2014), 589–680 | DOI | DOI | Zbl

[14] J. Kigami, “Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees”, Adv. Math., 225:5 (2010), 2674–2730 | DOI | MR | Zbl

[15] A. Bendikov, P. Krupski, “On the spectrum of the hierarchical Laplacian”, Potential Anal., 41:4 (2014), 1247–1266 | DOI | MR | Zbl

[16] V. S. Vladimirov, “Generalized functions over the field of $p$-adic numbers”, Russian Math. Surveys, 43:5 (1988), 19–64 | DOI | MR | Zbl

[17] V. S. Vladimirov, I. V. Volovich, “$p$-adic Schrödinger-type equation”, Lett. Math. Phys., 18:1 (1989), 43–53 | DOI | MR | Zbl

[18] V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-adic analysis and mathematical physics, Ser. Soviet East European Math., 1, World Sci. Publ., River Edge, NJ, 1994, xx+319 pp. | MR | MR | Zbl | Zbl

[19] E. Lukacs, Characteristic functions, 2nd ed., Hafner Publishing Co., New York, 1970, x+350 pp. | MR | MR | Zbl | Zbl

[20] C. C. Graham, O. C. McGehee, Essays in commutative harmonic analysis, Grundlehren Math. Wiss., 238, Springer-Verlag, New York–Berlin, 1979, xxi+464 pp. | MR | Zbl

[21] B. Solomyak, “Notes on Bernoulli convolutions”, Fractal geometry and applications: a jubilee of B. Mandelbrot, Proc. Sympos. Pure Math., 72, Part 1, Amer. Math. Soc., Providence, RI, 2004, 207–230 | MR | Zbl

[22] Y. Peres, B. Solomyak, “Absolute continuity of Bernoulli convolutions, a simple proof”, Math. Res. Lett., 3:2 (1996), 231–239 | DOI | MR | Zbl

[23] Y. Peres, W. Schlag, B. Solomyak, “Sixty years of Bernoulli convolutions”, Fractal geometry and stochastics (Greifswald/Koserow, 1998), v. II, Progr. Probab., 46, Birkhäuser, Basel, 2000, 39–65 | MR | Zbl

[24] M. Del Muto, A. Figà-Talamanca, “Diffusion on locally compact ultrametric spaces”, Expo. Math., 22:3 (2004), 197–211 | DOI | MR | Zbl

[25] M. Del Muto, A. Figà-Talamanca, “Anisotropic diffusion on totally disconnected abelian groups”, Pacific J. Math., 225:2 (2006), 221–229 | DOI | MR | Zbl

[26] P. Cartier, “Fonctions harmoniques sur un arbre”, Convegno di Calcolo delle Probabilità, INDAM (Rome, 1971), Sympos. Math., IX, Academic Press, London, 1972, 203–270 | MR | Zbl

[27] W. Woess, Denumerable Markov chains. Generating functions, boundary theory, random walks on trees, EMS Textbk. Math., Eur. Math. Soc., Zürich, 2009, xviii+351 pp. | DOI | MR | Zbl

[28] M. H. Taibleson, Fourier analysis on local fields, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo, 1975, xii+294 pp. | MR | Zbl

[29] A. N. Kochubei, Pseudo-differential equations and stochastics over non-Archimedian fields, Monogr. Textbooks Pure Appl. Math., 244, Marcel Dekker, Inc., New York, 2001, xii+316 pp. | DOI | MR | Zbl

[30] I. A. Ibragimov, Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters–Noordhoff Publishing Co., Groningen, 1971, 443 pp. | MR | MR | Zbl | Zbl

[31] B. Solomyak, “On the random series $\sum \pm \lambda ^{n}$ (an Erdös problem)”, Ann. of Math. (2), 142:3 (1995), 611–625 | DOI | MR | Zbl

[32] M. R. Leadbetter, G. Lindgren, H. Rootzén, Extremes and related properties of random sequences and processes, Springer Ser. Statist., Springer-Verlag, New York–Berlin, 1983, xii+336 pp. | MR | Zbl

[33] S. Chatterjee, P. Diaconis, E. Meckes, “Exchangeable pairs and Poisson approximation”, Probab. Surv., 2 (2005), 64–106 | DOI | MR | Zbl

[34] R. Arratia, L. Goldstein, L. Gordon, “Two moments suffice for Poisson approximations: the Chen–Stein method”, Ann. Probab., 17:1 (1989), 9–25 | DOI | MR | Zbl