On Bernoulli decomposition of random variables and recent various applications

Germinet, François

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2007-2008), Exposé no. 9, 12 p.

In this review, we first recall a recent Bernoulli decomposition of any given non trivial real random variable. While our main motivation is a proof of universal occurence of Anderson localization in continuum random Schrödinger operators, we review other applications like Sperner theory of antichains, anticoncentration bounds of some functions of random variables, as well as singularity of random matrices.

[A] Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1163-1182 (1994) | MR | Zbl

[AENSSt] Aizenman, M., Elgart, A., Naboko, S., Schenker, J., Stolz, G.: Moment analysis for localization in random Schrödinger operators. Inv. Math. 163, 343-413 (2006) | MR | Zbl

[AGKW] Aizenman, M., Germinet, F., Klein, A., Warzel, S.: On Bernoulli decompositions for random variables, concentration bounds and spectral localization, to appear in Prob. Th. Rel. F.

[AM] Aizenman, M., Molchanov, S.: Localization at large disorder and extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245-278 (1993) | MR | Zbl

[An] Anderson, I.: Combinatorics of finite sets. Corrected reprint of the 1989 edition. Dover Publications Inc. Mineola, NY, 2002. | MR | Zbl

[B] B. Bollobas, Random graphs, Academic Press 1985. | MR | Zbl

[BoK] Bourgain, J., Kenig, C.: On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161, 389-426 (2005) | MR | Zbl

[BrG] Bruneau, L., Germinet, F.: On the singularity of random matrices with independent entries. To appear in Proc. AMS.

[CH1] Combes, J.M., Hislop, P.D.: Localization for some continuous, random Hamiltonians in d-dimension. J. Funct. Anal. 124, 149-180 (1994) | MR | Zbl

[CHKl1] Combes, J.M., Hislop, P.D., Klopp, F.: Hölder continuity of the integrated density of states for some random operators at all energies. IMRN 4, 179-209 (2003) | MR | Zbl

[CHKl2] Combes, J.M., Hislop, P.D., Klopp, F.: Optimal Wegner estimate. Duke Math. J. 140, 469-498 (2007) | MR | Zbl

[CHKlNa] Combes, J.M., Hislop, P.D., Klopp, F. Nakamura, S.: The Wegner estimate and the integrated density of states for some random operators. Spectral and inverse spectral theory (Goa, 2000), Proc. Indian Acad. Sci. Math. Sci. 112 , 31-53 (2002) | MR | Zbl

[CHKlR] Combes, J.M., Hislop, P.D., Klopp, F, Raikov, G..: Global continuity of the integrated density of states for random Landau Hamiltonians. Comm. Partial Differential Equations. 29, 1187-1213 (2004) | MR | Zbl

[CHNa] Combes, J.M., Hislop, P.D., Nakamura, S.: The ${\mathrm{L}}^p$-theory of the spectral shift function, the Wegner estimate and the integrated density of states for some random operators. Commun. Math. Phys. 218, 113-130 (2001) | MR | Zbl

[DSiSt] Damanik, D., Sims, R., Stolz, G.: Localization for one dimensional, continuum, Bernoulli-Anderson models. Duke Math. J. 114, 59-100 (2002) | MR | Zbl

[DSto] Damanik, D., Stollmann, P.: Multi-scale analysis implies strong dynamical localization. Geom. Funct. Anal. 11, 11-29 (2001) | MR | Zbl

[DeRJLSi] Del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum IV: Hausdorff dimensions, rank one pertubations and localization. J. d’Analyse Math. 69, 153-200 (1996) | Zbl

[Do] Doeblin, W.: Sur les sommes d’un grand nombre de variables aléatoires indépendantes, Bull. Sci. Math. 63 23–64 (1939).

[DoL] Doeblin, W, Lévy, P.: Calcul des probabilités. Sur les sommes de variables aléatoires indépendantes à dispersions bornées inférieurement, C.R. Acad. Sci. 202 2027-2029 (1936).

[DrK] von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285-299 (1989) | MR | Zbl

[En1] Engel, K.: Optimal representations of partially ordered sets and a limit Sperner theorem. European J. Combin. 7, 287-302 (1986) | MR | Zbl

[En2] Engel, K.: Sperner theory. Encyclopedia of Mathematics and its Applications 65. Cambridge University Press, Cambridge, 1997. | MR | Zbl

[Er] Erdös, P.: On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51, 898–902 (1945). | MR | Zbl

[Es] Esseen, C. G.: On the concentration function of a sum of independent random variables Z. Wahrscheinlichkeitstheorie verw. Geb. 9, 290-308 (1968). | MR | Zbl

[FrMaScSp] Fröhlich, J.: Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101, 21-46 (1985) | MR | Zbl

[FrSp] Fröhlich, J., Spencer, T.: Absence of diffusion with Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151-184 (1983) | MR | Zbl

[G1] Germinet, F.: Dynamical localization II with an application to the almost Mathieu operator. J. Stat Phys. 95, 273-286 (1999) | MR | Zbl

[G2] Germinet, F.: Recent advances about localization in continuum random Schrödinger operators with an extension to underlying Delone sets. Proceedings of Qmath10 conference, Moeciu 2007. To appear.

[GDB] Germinet, F., De Bièvre, S.: Dynamical localization for discrete and continuous random Schrödinger operators. Commun. Math. Phys. 194, 323-341 (1998) | MR | Zbl

[GHK1] Germinet, F., Hislop, P., Klein, A.: On localization for the Schrödinger operator with a Poisson random potential. C.R. Acad. Sci. Paris Ser. I 341, 525-528 (2005) | MR | Zbl

[GHK2] Germinet, F., Hislop, P., Klein, A.: Localization for the Schrödinger operator with a Poisson random potential. J. Europ. Math. Soc. 9, 577-607 (2007) | MR

[GHK3] Germinet, F., Hislop, P., Klein, A.: Localization at low energies for attractive Poisson random Schrödinger operators. Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov, CRM Proceedings and Lecture Notes 42, 2007 | MR

[GK1] Germinet, F., Klein, A.: Bootstrap Multiscale Analysis and Localization in Random Media. Commun. Math. Phys. 222, 415-448 (2001) | MR | Zbl

[GK2] Germinet, F., Klein, A.: New characterizations of the region of complete localization for random Schrödinger operators. J. Stat. Phys. 122, 73-94 (2006) | MR | Zbl

[GK3] Germinet, F., Klein, A.: Localization for some Cantor-Anderson Schrödinger operators. Adventures in Mathematical Physics, Contemp. Math., vol. 447, 2007

[GK4] Germinet, F., Klein, A.: Universal occurence of localization for some continuum random Schrödinger operators. In preparation.

[GKS] Germinet, F, Klein, A., Schenker, J.: Dynamical delocalization in random Landau Hamiltonians. Annals of Math. 166, 215-244 (2007). | MR

[GoMP] Gol’dsheid, Ya., Molchanov, S., Pastur, L.: Pure point spectrum of stochastic one dimensional Schrödinger operators. Funct. Anal. Appl. 11, 1–10 (1977). | Zbl

[HuKiNaStoV] Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veselić, I.: Bounds on the spectral shift function and the density of states. Comm. Math. Phys. 262, 489-503 (2006). | MR | Zbl

[I] Ignatiouk-Robert, I.: Martin boundary of a killed random walk on a half-space, J. th. prob. 21, (2008). | MR

[KKoS] J. Kahn, J. Komlós, E. Szemerédi, On the probability that a random $\pm 1$ matrix is singular, J. Amer. Math. Soc. 8, 223-240 (1995). | MR | Zbl

[Ke] Kesten, H.: A sharper form of the Doeblin-Lévy-Kolmogorov-Rogozin inequality for concentration functions. Math. Scand. 25, 133-144 (1969). | MR | Zbl

[KiMa] Kirsch, W., Martinelli, F.: On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math. 334, 141-156 (1982) | MR | Zbl

[KiStoSt] Kirsch, W., Stollman, P., Stolz, G.: Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stochastic Equations 6, 241-268 (1998) | MR | Zbl

[K] Klein, A.: Multiscale analysis and localization of random operators. In Random Schrodinger operators: methods, results, and perspectives. Panorama & Synthèse, Société Mathématique de France. To appear.

[Kl2] Klopp, F.: Localization for continuous random Schrödinger operators. Commun. Math. Phys. 167, 553-569 (1995) | MR | Zbl

[Kl3] Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators. Duke Math. J. 98, 335-396 (1999) | MR | Zbl

[Kl4] Klopp F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Ann. Henri Poincaré 3, 711-737 (2002) | MR | Zbl

[Ko] Kolmogorov, A. N.: Sur les propriétés des fonctions de concentration de M. P. Lévy. Ann. Inst. H. Poincaré 16, 27-34 (1958-60). | Numdam | MR | Zbl

[Kom1] J. Komlós, On the determinant of $(0,1)$ matrices, Studia. Sci. Math. Hungar. 2, 7-22 (1967). | MR | Zbl

[Kom2] J. Komlós, On the determinant of random matrices, Studia. Sci. Math. Hungar. 3, 387-399 (1968). | MR | Zbl

[KotSi] Kotani, S., Simon, B.: Localization in general one-dimensional random systems. II. Continuum Schrödinger operators. Comm. Math. Phys. 112, 103-119 (1987) | MR | Zbl

[LeMuW] Leschke, H., Müller, P., Warzel, S.: A survey of rigorous results on random Schrödinger operators for amorphous solids. Markov Process. Related Fields 9, 729-760 (2003) | MR

[MaSc] Martinelli, F., Scoppola, E.: Introduction to the mathematical theory of Anderson localization. Riv. Nuovo Cimento 10, 1-90 (1987) | MR

[McD] McDonald, D.: A local limit theorem for large deviations of sums of independent, nonidentically distributed random variables. Ann. Probab. 7, 526–531 (1979). | MR | Zbl

[R1] Rogozin, B. A.: An Estimate for Concentration Functions, Theory Probab. Appl., 6, 94-97 (1961) Russian original: Teoriya Veroyatnost. i Prilozhen. 6 103–105 (1961). | MR | Zbl

[R2] Rogozin, B. A.: An integral-type estimate for concentration functions of sums of independent random variables Dokl. Akad. Nauk SSSR 211, 1067-1070 (1973) [in Russian]. | MR | Zbl

[R3] Rogozin, B. A.: Concentration function of a random variable. In Encyclopaedia of Mathematics, M. Hazewinkel (ed.), Springer 2002.

[Sl] A. Slinko, A generalization of Komlós’s theorem on random matrices, New Zealand J. Math. 30 no. 1, 81-86 (2001). | Zbl

[Sp] Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Zeit. 27, 544-548 (1928). | MR

[Sto1] Stollmann, P.: Wegner estimates and localization for continuum Anderson models with some singular distributions. Arch. Math. (Basel) 75, 307-311 (2000) | MR | Zbl

[Sto2] Stollmann, P.: Caught by disorder: bound states in random media. Boston: Birkhäusser, 2001. | MR | Zbl

[TV1] T. Tao, V. Vu, On Random $\pm 1$ matrices: Singularity and Determinant, Random Structures and Algorithms 28, 1-23 (2006). | MR | Zbl

[TV2] T. Tao, V. Vu, On the singularity of random Bernoulli matrices, J. Amer. Math. Soc. 20, 603-628 (2007). | MR | Zbl