Random walks on SL_2(C): spectral gap and limit theorems

We obtain various new limit theorems for random walks on SL2(C) under low moment conditions. For non-elementary measures with a finite second moment we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of É. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate O(1/ √ n) for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space W 1,2 that provides uniform estimates. —— In memory of Professor Nessim Sibony ——

[1]  M. Mirzakhani,et al.  Introduction to Ergodic theory , 2010 .

[2]  Emile Le Page,et al.  Théorèmes limites pour les produits de matrices aléatoires , 1982 .

[3]  Conditioned local limit theorems for random walks defined on finite Markov chains , 2017, Probability Theory and Related Fields.

[4]  B. M. Fulk.  MATH , 1992 .

[5]  Super-potentials of positive closed currents, intersection theory and dynamics , 2007, math/0703702.

[6]  J. Conze,et al.  Densité d’orbites d’actions de groupes linéaires et propriétés d’équidistribution de marches aléatoires , 2002 .

[7]  J. A. Salvato.  John wiley & sons. , 1994, Environmental science & technology.

[8]  H. Furstenberg.  Noncommuting random products , 1963 .

[9]  J. Dedecker,et al.  On the Komlós, Major and Tusnády strong approximation for some classes of random iterates , 2017, 1706.08282.

[10]  Berry–Esseen bound and precise moderate deviations for products of random matrices , 2019, Journal of the European Mathematical Society.

[11]  T. Dinh,et al.  Dynamics of holomorphic correspondences on Riemann surfaces , 2018, International Journal of Mathematics.

[12]  Decrease of Fourier coefficients of stationary measures , 2017, Mathematische Annalen.

[13]  Gabriel Vigny.  Dirichlet-like space and capacity in complex analysis in several variables , 2007, 0705.3164.

[14]  Yves Benoist,et al.  Random Walks on Reductive Groups , 2016 .

[15]  Decay of correlations and the central limit theorem for meromorphic maps , 2004, math/0410008.

[16]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[17]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[18]  Hao Wu,et al.  Berry-Esseen bound and Local Limit Theorem for the coefficients of products of random matrices , 2021 .

[19]  BERRY-ESSEEN BOUNDS AND MODERATE DEVIATIONS FOR THE RANDOM WALK ON GL_d (R) , 2020 .

[20]  I. Grama,et al.  A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group , 2020, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[21]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[22]  J. Dedecker,et al.  LIMIT THEOREMS FOR THE LEFT RANDOM WALK ON GLd (R) , 2016, 1603.02217.

[23]  Tien-Cuong Dinh,et al.  Decay of Fourier coefficients for Furstenberg measures , 2021 .

[24]  Yves Benoist,et al.  Central limit theorem for linear groups , 2016 .

[25]  B. Lautrup,et al.  Products of random matrices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Tien-Cuong Dinh,et al.  Products of random matrices: a dynamical point of view , 2021, Pure and Applied Mathematics Quarterly.

[27]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[28]  S. Gouëzel.  Limit theorems in dynamical systems using the spectral method , 2015 .

[29]  Jialun Li.  Fourier decay, renewal theorem and spectral gaps for random walks on split semisimple Lie groups , 2018, Annales scientifiques de l'École Normale Supérieure.

[30]  C Cuny,et al.  Berry-Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GLd(R) , 2021 .

[31]  P. Bougerol,et al.  Products of Random Matrices with Applications to Schrödinger Operators , 1985 .