Maximum of Dyson Brownian motion and non-colliding systems with a boundary
Alexei Borodin, Patrik L. Ferrari, Michael Praehofer, Tomohiro Sasamoto, Jon Warren
证明了GUE Dyson的布朗运动的最大值与有壁非碰撞系统的一个等价律。这推广了布朗运动和反射布朗运动的最大值之间的公知关系。
We prove an equality-in-law relating the maximum of GUE Dyson's Brownian motion and the non-colliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion.
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We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson Brownian motions, we observe a decoupling phenomenon in the limit: the global interactions become negligible and only the local interactions remain. The resulting limiting objects are interacting particle systems which can be described as Brownian versions of certain totally asymmetric exclusion processes. This is the first appearance of a particle system with loc…
Non-colliding system of Brownian particles as Pfaffian process
文[7]研究了非碰撞布朗运动的时间非齐次系统，证明了多时间相关函数一般由Dyson和Mehta意义下的四元数行列式给出.本文用弗雷德霍姆行列式和FredholmPfronan给出了等价命题的另一个证明，并指出本系统是Pfronan过程的一个典型例子.
We study the linear statistics of the circular $β$-ensemble with a Stein's method argument, where the exchangeable pair is generated through circular Dyson Brownian motion. This generalizes previous results obtained in such a way for the CUE and provides a novel approach for studying linear statistics of $β$-ensembles. This approach allows studying simultaneously a collection of linear statistics whose number grows with the dimension of the ensemble. Also this approach requires estimating only low order moments of the linear statistics.
We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times $t=o(1)$ with deterministic initial data V . Our main result states that if the density of states of $V$ is bounded both above and away from $0$ down to scales $\ell \ll t$ in a small interval of size $G \gg t$ around an energy $E_0$, then the local statistics coincide with the GOE/GUE near the energy $E_0$ after time $t$. Our methods are partly based on the idea of coupling two Dyson Brownian motions from [6], the parabolic regularity result of [15], and the eigenvalue rigidity r…
Non-colliding Brownian Motions and the extended tacnode process
考虑两个起点两个端点的非碰撞布朗运动。选择这些点，使得两组布朗运动刚好接触，这种情况被称为接触点。本文用新的方法计算了行列式点过程在tacnode点的扩展核，并给出了与Delvaux，Kuijlaars和Zhang在以前的工作中单次得到的核不同的形式.扩展核的形式也不同于Adler、Ferrari和van Moerbeke在另一个模型中获得的扩展tacnode核。我们还得到了有限个非碰撞的布朗运动的相关核从两个点开始，并在任意点结束。
Multilevel Dyson Brownian motions via Jack polynomials
我们引入了任意参数$β>0$的Dyson布朗运动的多级形式，推广了$β=2$的Warren交错反射布朗运动.这样的过程统一$β$角落过程和戴森布朗运动在一个单一的对象。我们的方法是基于一定的多级离散马尔可夫链的独立利益，这是通过杰克对称多项式定义的近似。特别是，这种方法允许显示，在一个多层次的戴森布朗运动的水平交织（至少对于$β\ge 1$），并明确给出相应的链接。
For general $β\geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is distributed as the Airy$_β$ random point field. We prove that the increments of the limiting process are locally Brownian. When $β>1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{ö}lder for any $r<1-(1+β)^{-1}$. Furthermore for all $β\geq 1$ we show that the limiting process solves an SDE in a weak sense. When $β=2…
Dyson Brownian motion on a Jordan curve
本文研究了在乔丹曲线上严格构造迪萨恩布朗运动，并探讨其基本性质及在不同物理极限下的行为。
We consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale $η_* \geq N^{-2/3}$ and no outliers, then after times $t \gg \sqrt{ η_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{\varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bour…
Infinite systems of non-colliding Brownian particles
研究了一维非碰撞布朗粒子。$N$布朗粒子在时间0从原点开始，然后它们直到有限时间$T$才彼此碰撞。利用自对偶四元数矩阵，推导了多时间相关函数的行列式表达式。我们考虑了无限粒子$N \to \infty$和无限时间间隔$T \to \infty$的标度极限。根据标度，证明了多时间相关函数的两个极限定理，这两个极限定理可以定义时间非齐次的无穷粒子系统。
We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $β$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.
The Dyson Brownian minor process
考虑$n\times n$厄米矩阵值随机过程$\{H_t\}_{t\geq 0}$，其中矩阵元素根据Ornstein-Uhlenbeck过程演化。众所周知，特征值执行所谓的Dyson布朗运动，即它们表现为以永不相交为条件的Ornstein-Uhlenbeck过程。本文不仅研究了全矩阵的特征值，而且研究了所有主子式的特征值。即，$H_t$左上角的$k\times k$的特征值。如果你把这个过程投影到一个类空路径上，它是一个行列式过程，我们计算核。这个内核包含了著名的GUE小内核，由Johanson-Nordenwald和Okounkov-Reshetikhin在2006年发现，以及Dyson布朗运动内核，由Forrester-Nagao在1998年发现。在此内核的批量缩放限制，它是可能的，以恢复一个时间依赖性的概括Boutillier的珠内核。
Non-colliding Brownian bridges and the asymmetric tacnode process
我们考虑从两点出发并返回到相同位置的非碰撞布朗桥。选择这些位置，使得在大量桥的限制下，两族桥刚好彼此接触形成一个tacnode。我们得到了临界点处的极限过程，即“非对称临界点过程”。这是一个行列式点过程，其相关核由两个参数给出：（1）两族桥梁极限形状的曲率比λ>0，（2）控制涨落尺度上相互作用的参数σ。这推广了对称tacnode过程的结果（λ=1的情况）。
The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $κ\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the…
Fixed energy universality for Dyson Brownian motion
本文证明经典Dyson布朗运动在固定能量意义下局部特征值统计与GOE/GUE一致，并推广到任意已知平均能量普适性的随机矩阵系综，方法基于均匀化理论。
We introduce a semi-implicit Euler-Maruyama approximation which preservers the non-colliding property for some class of non-colliding particle systems such as Dyson Brownian motions, Dyson-Ornstein-Uhlenbeck processes and Brownian particles systems with nearest neighbour repulsion, and study its rates of convergence in both $L^p$-norm and path-wise sense.
Dyson's Brownian motions, intertwining and interlacing
利用一族反射布朗运动构造了非碰撞布朗运动的Dyson过程。给出了一些显式公式，包括一个家庭的合并布朗运动的分布。
Global Fluctuations in General Beta Dyson Brownian Motion
本文考虑了在二次外势作用下，具有排斥静电相互作用的实直线上的扩散粒子系统。已知当粒子数趋于无穷大时，经验测度过程弱收敛到确定性测度值过程。假设初始波动很小，经验测量过程的重新标度的线性统计在分布上收敛到足够光滑的检验函数的高斯极限。通过分析描述极限涨落的偏微分方程，我们导出了中心极限定理中均值和协方差的显式一般公式。
On mesoscopic equilibrium for linear statistics in Dyson's Brownian Motion
本文用$β=2$研究Dyson布朗运动的介观涨落。戴森证明了高斯酉熵（GUE）是这种随机演化的不变测度，并指出，当从初始点的一般配置开始时，GUE统计量成为主导所需的时间取决于我们所观察的尺度：微观相关性比宏观相关性更快地到达平衡状态。在本文中，我们调查过渡的中间，即介观尺度。我们考虑的时间尺度是这样的，即系统已经处于微观平衡（局部关联的正弦普适性），但我们还没有达到宏观尺度的平衡。我们描述过渡到平衡的所有介观尺度上的中心极限定理的线性统计足够光滑的测试功能。我们考虑两种情况：确定性的初始点和随机选择的初始点。在随机情形下，我们得到了一个从经典的独立随机变量中心极限定理到GUE中心极限定理的过渡。
We show that for all positive beta the semigroups of beta-Dyson Brownian motions of different dimensions are intertwined. The proof relates beta-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption beta>1 in [GS]. The corresponding results for beta-Dyson Ornstein-Uhlenbeck processes are also presented.
In this article, we study multiple $SLE_κ$, for $κ\in(0,4]$, driven by Dyson Brownian motion. This model was introduced in the unit disk by Cardy in connection with the Calogero-Sutherland model. We prove the Carathéodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity $κ\in (0,4]$ for the case of $N=2$ driving forces. Our proofs use the analysis of Bessel processes and estimates on Loewner differential equation with multiple driving forces. In the last section, we estimate the Hausdorff distance of the hulls under perturbations of the…
Quasi-stationarity of the Dyson Brownian motion with collisions
本文研究了受碰撞影响的Dyson布朗运动在退出状态空间子域前的准稳态行为，通过多值SDE理论和Krein-Rutman定理分析了其存在性和唯一性。
We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N}$, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are…
We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the $β$-Dyson's Brownian motion. For $β\in\{1,2,4\}$ this is a consequence of the Gaussian case, however the relation holds for general $β$. We further raise the question whether there is an analogue of $β$-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case $n=2$ we give a simple construction.
We prove tightness and limiting Brownian-Gibbs description for line ensembles of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. Statistical properties of the resulting ensemble are very different from that of non-colliding Brownian bridges without self-potentials. The model itself was introduced in order to mimic level lines of (2+1)-dimensional discrete Solid-On-Solid random interfaces above a hard wall.
Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)
本文证明了圆上Dyson布朗运动在耦合参数β趋于无穷时的有限时间大偏差原理，刻画了其零能量与有限能量系统的长时间行为，并将结果推广到κ趋于0+时多径向SLE的有限时间大偏差。
Support of Dyson Brownian Motion
该研究针对β≥1的β-戴森布朗运动，通过确定性局部预解式排除原理，证明了粒子在指定时间内保持在半圆自由卷积流支撑集的ε邻域内的高概率结论，适用于多分支支撑集且无需初始谱边界正则性假设。
The Infinite Dyson Brownian Motion with $β=2$ Does Not Have a Spectral Gap
本文证明了逆温度 $β=2$ 的未标记无穷 Dyson 布朗运动对应的 Dirichlet 形式不存在谱隙，揭示了该过程的谱性质。
Applications of optimal transport to Dyson Brownian Motions and beyond
本研究基于最优传输中的Caffarelli收缩定理提出新方法，得到β≥2时β-戴森布朗运动的精确一致连续模估计，并将方法推广至多种对数凹扰动型随机曲线集合。
Complex Brownian Motion Representation of the Dyson Model
Dyson的布朗运动模型（含参数$β=2$），本文简称Dyson模型，它是A型Weyl腔中吸收布朗运动的$h$变换。根据有限个粒子的初始构型，定义了一组整函数，并引入了独立复布朗运动（CBM）系统的鞅，该鞅由一个矩阵的行列式表示，该矩阵的元素由CBM通过整函数的共形变换给出.我们证明了Dyson模型可以用这个行列式鞅加权的独立CBM系统来表示。从这个CBM表示，Eynard-Mehta型相关核的推导和戴森模型被证明是决定性的。CBM表示是$h$ -变换的一个有用的扩展，因为它也适用于无穷粒子系统。利用这种表示，我们证明了一系列的过程，收敛到戴森模型的无限数量的粒子，和非碰撞的极限过程的性质的紧密性。
Dyson's non-intersecting Brownian motions with a few outliers
考虑实数线上的n个不相交粒子（戴森布朗运动），所有粒子在时间=0时从原点开始，并在时间=1时被迫返回x=0。对于大的n，粒子的平均密度的平均值有它的支持，对于每个0<t<1，在椭圆的内部。艾里过程被定义为这些不相交的布朗运动的运动，但从椭圆上的任意一点与适当的时空重新标度。现在假设这些粒子中的有限个r被迫到不同的目标点。当n较大时，它是否影响椭圆上的布朗涨落？在本文中，我们表明，没有新的过程出现，只要一个考虑点的椭圆，其t坐标小于t坐标的切点的切线曲线通过目标点。在这个切点上，波动服从一个新的统计：具有r个异常值的艾里过程（简称：{ \bf r-Airy过程}）。这个新过程的转移概率的对数由一个新核（扩展了Airy核）的弗雷德霍姆行列式给出，它满足关于x和时间的非线性偏微分方程.
We study the spectrum of the kinetic Brownian motion in the space of $d\times d$ Hermitian matrices, $d\geq2$. We show that the eigenvalues stay distinct for all times, and that the process $Λ$ of eigenvalues is a kinetic diffusion (i.e. the pair $(Λ,\dotΛ)$ of $Λ$ and its derivative is Markovian) if and only if $d=2$. In the large scale and large time limit, we show that $Λ$ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.
This paper is the third chapter of three of the author's undergraduate thesis. In this paper, we study the convergence of local bulk statistics for linearized covariance matrices under Dyson's Brownian motion. We consider deterministic initial data $V$ approximate the Dyson Brownian motion for linearized covariance matrices by the Wigner flow. Using universality results for the Wigner flow, we deduce universality for the linearized covariance matrices. We deduce bulk universality of averaged bulk correlation functions for both biregular bipartite graphs and honest covariance matrices. We also …
Dyson Brownian Motion and motion by mean curvature
通过将Riemannian流形上Brownian motion的外在构造适应于Hermitian矩阵空间的群轨道几何，构造了$\beta\in (0,\infty]$的Dyson Brownian motion，并揭示了$\beta$无穷大时特征值与群轨道的演化规律。
Consecutive Minors for Dyson's Brownian Motions
1962年，戴森在随机矩阵模型中引入了动力学，特别是GUE（也适用于beta=1和4），让条目根据独立的Ornstein-Uhlenbeck过程演化。Dyson证明了矩阵的谱点是按照不相交的布朗运动演化的。本文证明了两个连续的主子式的交错谱也构成一个马尔可夫过程（扩散）。这种扩散由两组戴森非交叉布朗运动组成，具有关于交织的特定相互作用。这以生成元、转移概率和不变测度的形式显示出来，在这里提供;这在所有情况下都是这样做的：beta=1，~2，~4。当β=2，~4时，三个连续子式的谱不再是马尔可夫的.
Quantum Separation of Variables and Multi-component Dyson Brownian Motion
本文定义了一种新的库仑气体，它是由两种物质的任意数量的点电荷组成的，它们在静电斥力的影响下作布朗运动。作为Dyson提出的全同粒子模型的推广，该模型是描述各种随机矩阵系综的非平衡态的动力学系统，我们的系统给出了$ Q_1 $和$ Q_2 $大小的电荷的布朗运动的精确数学描述，使得$ βQ_1 Q_2 = 1 $在直线或圆上，其中$β$是气体温度的倒数.
From maximal entropy exclusion process to unitary Dyson Brownian motion and free unitary hydrodynamics
该研究通过分析最大熵排除过程，揭示了其在低密度和流体动力学极限下与幺正Dyson布朗运动和自由幺正流体动力学之间的联系，利用Schur多项式和特征标理论作为核心工具。
A new approach for the unitary Dyson Brownian motion through the theory of viscosity solutions
本文通过粘度解理论研究单位元Dyson布朗运动，证明其极限谱测度的原函数是粘度方程的唯一解，并探讨了解的正则化和长时间行为。
We prove that self-diffusion constants of interacting Brownian particles in $ \mathbb{R}$ always vanish if the particles do not collide with each other. We represent self-diffusion constants by additive functionals of reversible Markov processes as obtained in O. 1998.
Multilevel Dyson Brownian motions via the superposition principle
本文利用Figalli-Trevisan叠加原理，统一构造了所有β>2的多级Dyson布朗运动，推广了其随机微分方程表示，证明唯一性，并确认β↓2时极限为Warren过程。
We study Dyson Brownian motion with general potential $V$ and for general $β\geq 1$. For short times $t = o (1)$ and under suitable conditions on $V$ we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $η_* \ll t \ll 1$, we prove a mesoscopic central limit theorem for linear statistics at all scales $N^{-1}\llη\ll t$
In this article, we consider flat and curved Riemannian symmetric spaces in the complex case and we study their basic integral kernels, in potential and spherical analysis: heat, Newton, Poisson kernels and spherical functions, i.e. the kernel of the spherical Fourier transform. We introduce and exploit a simple new method of construction of these $W$-invariant kernels by alternating sum formulas. We then use the alternating sum representation of these kernels to obtain their asymptotic behavior. We apply our results to the Dyson Brownian Motion on $R^d$.
A phase transition in the Bakry-Émery gradient estimate for Dyson Brownian motion
本文证明有限粒子 Dyson 布朗运动在逆温度 $0<β<1$ 时，Bakry-Émery $N$-Ricci 张量下界非负但梯度估计 ${\sf BE}(0,N)$ 不成立，揭示了小逆温度区域的曲率界相变现象。
On the Limit of the Tridiagonal Model for $β$-Dyson Brownian Motion
本文研究了β-戴逊布朗运动下三对角模型的极限，推导了动态β-随机Airy算子的结构，并通过数值验证了β=1,2,4的情况。
In this paper, we compare the solutions of Dyson Brownian motion with general $β$ and potential $V$ and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and potential $V$, we obtain the optimal rigidity estimates of particle locations near the edge for short time $t=\text{o}(1)$. Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge which, as far as we know, have been done f…
Multiple SLEs and Dyson Brownian motion: transition density and Green's function
本文研究 $κ\in(0,4]$ 的多重弦 SLE，证明在公共时间参数化下其驱动函数的转移密度可由 Dyson 布朗运动的转移密度与 Green 函数表示。
We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of $2+1$-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari--Spohn diffusions associated with limiting Sturm--Liouville operators. In particular, the limiting invariant measures are given by the squar…
We compute analytically the probability $S(t)$ that a set of $N$ Brownian paths do not cross each other and stay below a moving boundary $g(τ)= W \sqrtτ$ up to time $t$. We show that for large $t$ it decays as a power law $S(t) \sim t^{- β(N,W)}$. The decay exponent $β(N,W)$ is obtained as the ground state energy of a quantum system of $N$ non-interacting fermions in a harmonic well in the presence of an infinite hard wall at position $W$. Explicit expressions for $β(N,W)$ are obtained in various limits of $N$ and $W$, in particular for large $N$ and large $W$. We obtain the joint distribution…
Limit theorems for moment processes of beta Dyson's Brownian motions and beta Laguerre processes
本文研究了参数beta与系统尺寸倒数成正比时，beta Dyson Brown运动和beta Laguerre过程矩过程的动态极限定理，建立了相应的LLN和CLT。
Resetting Dyson Brownian motion
本文提出重置 Dyson 布朗运动（β-RDBM），研究随机重置驱动下 Dyson 对数气体从平衡态到非平衡稳态的转变，精确求解稳态联合分布并揭示其涨落由刚性变为蓬松的机制。
We consider tightness for families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of $2+1$ discrete Solid-On-Solid random interfaces above a hard wall.
Dyson's model in infinite dimensions is a system of Brownian particles interacting via a logarithmic potential with an inverse temperature of $ β= 2$. The stochastic process is given as a solution to an infinite-dimensional stochastic differential equation. Additionally, a Dirichlet form with the sine$ _2$ point process as a reference measure constructs the stochastic process as a functional of the associated configuration-valued diffusion process. In this paper, we prove that Dyson's model in infinite dimensions is irreducible.
In this paper, we use Malliavin calculus to show the existence and continuity of density functions of $d$-dimensional non-colliding particle systems such as hyperbolic particle systems and Dyson Brownian motion with smooth drift. For this purpose, we apply results proved by Florit and Nualart (1995) and Naganuma (2013) on locally non-degenerate Wiener functionals.
In this paper, we consider tridiagonal matrices the eigenvalues of which evolve according to $β$-Dyson Brownian motion. This is the stochastic gradient flow on $\mathbb{R}^n$ given by, for all $1 \leq i \leq n,$ \[ dλ_{i,t} = \sqrt{\frac{2}β}dZ_{i,t} - \biggl( \frac{V'(λ_i)}{2} - \sum_{j: j \neq i} \frac{1}{λ_i - λ_j} \biggr)\,dt \] where $V$ is a constraining potential and $\left\{ Z_{i,t} \right\}_1^n$ are independent standard Brownian motions. This flow is stationary with respect to the distribution \[ ρ^β_N(λ) = \frac{1}{Z^β_N} e^{-\fracβ{2} \left( -\sum_{1 \leq i \neq j \leq N} \log|λ_i -…
Edge homogenization of Dyson Brownian motion and applications
证明了耦合Dyson Brownian运动在谱边缘的均化估计，并应用于量化广义Wigner矩阵特征值分布的边缘普适性，给出了Kolmogorov-Smirnov距离的上下界。
Edge Rigidity of Dyson Brownian Motion with General Initial Data
本文在仅要求平均初始密度具有平方根剖面下界的条件下，证明一般 β 的 Dyson 布朗运动极端粒子涨落界为 (log n)^{O(1)}n^{-2/3}，并由此得到短时间内 Tracy-Widom β 分布的收敛性。
Quantum cohomology of the Grassmannian and unitary Dyson Brownian motion
本文研究一类交换马尔可夫核，其描述离散圆上不相交粒子的运动，并关联格拉斯曼流形的量子上同调环；在k固定时，对约n^2个核的乘积证明了Berry-Esseen定理和局部极限定理，进而得到量子上同调环以SU(k)热核表示的渐近公式。
We study the generalized Dyson Brownian motion (GDBM) of an interacting $N$-particle system with logarithmic Coulomb interaction and general potential $V$. Under reasonable condition on $V$, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $\mathcal {C}([0,T],\mathscr{P}(\mathbb{R}))$ and all the large $N$ limits satisfy a nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, we prove that the McKean-Vlasov equation is indeed the g…
What is a $p$-adic Dyson Brownian motion?
本文分类并构造 $\mathrm{GL}_N(\mathbb{Q}_p)$ 上的连续时间 Markov 跳过程，证明其奇异数构成带 Weyl 腔壁反射的 Poisson 跳过程，从而给出乘法 Dyson Brownian 运动的 $p$ 进类比。
We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the a…
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