Central‐limit theorems on groups

摘要: The probability density pN of the product N statistically independent (and identically distributed, each with p1) elements a group is studied in limit N→∞. It shown, for compact groups R(2) and R(3), that pN→1 as N→∞, independently p1. made plausible similar behavior to be expected other groups. For noncompact groups, case SU(1,1)which interest physics disordered conductors, studied. which p1 isotropic, i.e., phases, analyzed detail. When fixed N≫1, Gaussian distribution appropriate variable found. original variables are rescaled by 1/N N→∞ taken, keeping ratio length conductor localization fixed, an explicit integral representation resulting also exhibited latter satisfies ‘‘diffusion’’ equation on manifold.