Conditional Haar measures on classical compact groups

经典紧致群集合的总体收敛速率

Rates of Bulk Convergence for Ensembles of Classical Compact Groups

本文研究了经典紧致群集合在不同群上的随机矩阵收敛速率，发现单位群收敛速率为$ N^{-2} $，正交群和辛群收敛速率为$ N^{-1} $。

Borel theorems for random matrices from the classical compact symmetric spaces

We study random vectors of the form $(\operatorname {Tr}(A^{(1)}V),...,\operatorname {Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^{(ν)}$ are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and Śniady, for polynomial functions on th…

Empirical central limit theorems for ergodic automorphisms of the torus

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f o T^i)_{i \geq 1} under some mild conditions on the modulus of continuity of f. The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural filtration.

紧李群随机元素幂的渐近性

Asymptotics of powers of random elements of compact Lie groups

本文研究紧李群中一般分布随机元素幂的渐近行为，在温和绝对连续性条件下证明其特征值分布收敛到 Haar 元素固定幂的分布，并进一步分析幂本身的极限分布。

Random walks in compact groups

Let X_1,X_2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product X_l*...*X_1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.

Integration formulas for Brownian motion on classical compact Lie groups

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur-Weyl dualities based on stochastic calculus.

Weakly infinitely divisible measures on some locally compact Abelian groups

On the torus group, on the group of p-adic integers and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of the real numbers. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.

Maximum Entropy on Compact Groups

On a compact group the Haar probability measure plays the role of uniform distribution. The entropy and rate distortion theory for this uniform distribution is studied. New results and simplified proofs on convergence of convolutions on compact groups are presented and they can be formulated as entropy increases to its maximum. Information theoretic techniques and Markov chains play a crucial role. The convergence results are also formulated via rate distortion functions. The rate of convergence is shown to be exponential.

Random variables with an invariant random shift in compact metrisable abelian groups

The main result of this paper states that for independent random variables $X, Y$ taking values in a compact metrisable abelian group, $X + Y$ has the same distribution as $X$, if and only if there exists a compact subgroup $A$ such that $P(Y\in A)=1$ and $X + a$ has the same distribution as $X$ for all $a\in A$. As a conclusion from the above it is shown that for independent random variables $X, Y$ such that $X+Y$ has the same distribution as $X$, $X+Y$ and $Y$ are also independent. It becomes also apparent that the distribution of $X$ is the Haar measure (uniform distribution) if for each op…

Limit theorems on locally compact Abelian groups

We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem, since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure can not have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate specia…

Expectation values of polynomials and moments on general compact Lie groups

We develop a powerful framework to calculate expectation values of polynomials and moments on compact Lie groups based on elementary representation-theoretic arguments and an integration by parts formula. In the setting of lattice gauge theory, we generalize expectation value formulas for products of Wilson loops by Chatterjee and Jafarov to arbitrary compact Lie groups, and study explicit examples for many classical compact Lie groups and the exceptional Lie group $G_2$. Extending classical results by Collins and Lévy, we use our framework to derive expectation value formulas of polynomials o…

紧群上的矩方法：Weingarten 演算及其应用

Moment Methods on compact groups: Weingarten calculus and its applications

本文综述紧群与紧量子群 Haar 测度矩计算的 Weingarten 演算，总结其函数理论性质及在随机矩阵、量子概率等领域的应用。