On a gateway between continuous and discrete Bessel and Laguerre processes

随机贝塞尔算子在高温下的谱

The spectrum of the stochastic Bessel operator at high temperature

研究随机贝塞尔算子在高温极限下的谱行为，证明其特征值点过程收敛到非平凡极限过程，并推导了最大特征值的大偏差渐进行为。

Freezing Limits for Beta-Cauchy Ensembles

Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $β$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $β$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing result…

相互作用Bessel过程的两个极限区

Two limiting regimes of interacting Bessel processes

我们考虑相互作用的贝塞尔过程，一个家庭的多粒子系统在一个维度的粒子演变为个人贝塞尔过程和排斥通过一个对数势。我们考虑这个家庭的两个主要参数：逆温度β和贝塞尔指数nu的两个限制制度。我们得到的时间尺度的稳态分布的情况下，β或nu是大的，但有限的过程。特别是，对于大β，我们表明，系统的稳态分布对应于随机矩阵的β-Laguerre系综的本征值分布。我们还估计了在这两种情况下的弛豫时间的稳定状态。我们发现，在冻结区域β->无穷大，粒子的标度最终位置被锁定在参数为nu-1/2的Laguerre多项式的零点的平方根上，而在区域nu->无穷大，我们证明了粒子的标度最终位置收敛到一个点.为了得到我们的结果，我们使用了Dunkl算子理论，特别是B型缠绕算子。我们推导出一个以前未知的表达式，该运营商和研究其行为在这两个限制制度。利用纠缠算子的这些极限形式，我们得到了过程的稳态分布、弛豫时间的估计和极限行为。

Central limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes are classified via associated root systems and positive multiplicity constants. They describe the dynamics of interacting particle systems of Calogero-Moser-Sutherland type. Recently, Andraus, Katori, and Miyashita derived some weak laws of large numbers for these processes for fixed positive times and multiplicities tending to infinity. In this paper we derive associated central limit theorems for the root systems of types A, B and D in an elementary way. In most cases, the limits will be normal distributions, but in the B-case there are freezing limits where dis…

Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices

Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $β$-Hermite and $β$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to\infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigen…

Functional central limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $β$-Hermite and $β$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $β$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cd…

The differential equations associated with Calogero-Moser-Sutherland particle models in the freezing regime

Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models and are related with $β$-Hermite and $β$-Laguerre ensembles. They depend on a root system and a multiplicity $k$. Recently, several limit theorems for $k\to\infty$ were derived where the limits depend on the solutions of associated ODEs in these freezing regimes. In this paper we study the solutions of these ODEs which are are singular on the boundaries of their domains. In particular we prove that for a start in arbitrary boundary points, the ODEs always admit unique solutions in their domains for $t>0$.