Brownian Motion on the Unitary Quantum Group: Construction and Cutoff
Jean Delhaye
本文在自由酉量子群上构造了布朗运动的类比，并计算其截断剖面，为量子群上的随机过程研究提供了新结果。
在本研究中，我们在自由酉量子群上构造了布朗运动的一个类比，并计算了其截断剖面。
In this study, we construct an analog of the Brownian motion on free unitary quantum groups and compute its cutoff profile.
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Brownian motion on reflection quantum groups. Construction and cutoff
研究构造自由反射量子群上的布朗运动并计算其截止特性
The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian …
We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.
The first part of this paper is devoted to the Brown measure of the product of the free unitary Brownian motion by an arbitrary free non negative operator. Our approach follows the one recently initiated by Driver-Hall-Kemp though there are substantial differences at the analytical side. In particular, the corresponding Hamiltonian system is completely solvable and the characteristic curve describing the support of the Brown measure has a non-constant (in time) argument. In the second part, we specialize our findings to the product of the free unitary Brownian motion by a free self-adjoint pro…
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Summing free unitary Brownian motions with applications to quantum information
受量子信息启发，研究k个独立酉布朗运动之和的动态随机态，证明大尺寸极限下其谱分布与对应自由Jacobi过程的关系，推广矩展开结果并推导相关非正规算子矩生成函数的偏微分方程及特征曲线。
In this paper, we derive explicit expressions for the moments and for the mixed moments of the compression of a free unitary Brownian motion by a free projection. While the moments of this non-normal operator are readily derived using analytical or combinatorial methods, we only succeeded to derive its mixed ones after solving a non-linear partial differential equation for their two-variables generating function. Nonetheless, the combinatorics of non crossing partitions lead to another expression of the lowest-order mixed moment. We shall also give some interest in odd alternating moments. In …
Let $Z_N$ be a Ginibre ensemble and let $A_N$ be a Hermitian random matrix independent from $Z_N$ such that $A_N$ converges in distribution to a self-adjoint random variable $x_0$. For each $t>0$, the random matrix $A_N+\sqrt{t}Z_N$ converges in $\ast$-distribution to $x_0+c_t$, where $c_t$ is the circular variable of variance $t$, free from $x_0$. We use the Hamilton-Jacobi method to compute the Brown measure $ρ_t$ of $x_0+c_t$. The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of $x_0+c_t$ is related to the subo…
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We obtain a description for the spectral distribution of the free Jacobi process for any initial pair of projections. This result relies on a study of the unitary operator $RU_tSU_t^*$ where $R,S$ are two symmetries and $U_t$ a free unitary Brownian motion, freely independent from $\{R,S\}$. In particular, for non-null traces of $R$ and $S$, we prove that the spectral measure of $RU_tSU_t^*$ possesses two atoms at $\pm1$ and an $L^\infty$-density on the unit circle $\mathbb{T}$, for every $t>0$. Next, via a Szegő type transform of this law, we obtain a full description of the spectral distribu…
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