The scaling limit of Fomin's identity for two paths in the plane

Fractal behavior of multivariate operator-self-similar stable random fields

We investigate the sample path regularity of multivariate operator-self-similar stable random fields with values in $\mathbb{R}^m$ given by a harmonizable representation. Such fields were introduced in [25] as a generalization of both operator-self-similar stochastic processes and operator scaling random fields and satisfy the scaling property $\{X(c^E t) : t \in \mathbb{R}^d \} \stackrel{\rm d}{=} \{c^D X(t) : t \in \mathbb{R}^d \}$, where $E$ is a real $d \times d$ matrix and $D$ is a real $m \times m$ matrix. This paper provides the first results concerning sample path properties of such fi…

路径图上量子游走的Walk/Zeta对应：由广义反射原理蕴含的前几何结构

Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle

本文在有限路径图上建立反射与Szegedy游走的局部Konno-Sato对应，通过广义反射原理得到zeta分解，并在对称与非对称情形下恢复经典公式及大质量狄拉克型极限。

Fluctuations of random Motzkin paths

It is known that after scaling a random Motzkin path converges to a Brownian excursion. We prove that the fluctuations of the counting processes of the ascent steps, the descent steps and the level steps converge jointly to linear combinations of two independent processes: a Brownian motion and a Brownian excursion. The proofs rely on the Laplace transforms and an integral representation based on an identity connecting non-crossing pair partitions and joint moments of an explicit non-homogeneous Markov process.

Universal scaling limits of the symplectic elliptic Ginibre ensemble

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are…

随机排列中线性大小的递增子序列与permutons的Robinson-Schensted表

Increasing subsequences of linear size in random permutations and the Robinson-Schensted tableaux of permutons

本文扩展Robinson-Schensted对应到permutons，证明其RS-表非平凡时，采样排列的递增子序列呈线性增长，并给出大偏差结果及偏微分方程。

关于 $\nabla\phi$ 界面模型的 Ray-Knight 定理及标度极限

A Ray-Knight theorem for $\nablaϕ$ interface models and scaling limits

本文为凸势 $\nabla\phi$ 界面模型建立随机游走占据时间与梯度场平方的 Ray-Knight 型恒等式，并证明三维重整化平方收敛到 Gaussian 自由场的 Wick 序平方。

Rescaling Ward identities in the random normal matrix model

We study existence and universality of scaling limits for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward's equation, which is an identity satisfied by the 1-point function.

双参数Littlewood恒等式与稳定自旋Hall--Littlewood对称函数的半空间Yang--Baxter随机场

Two-parameter Littlewood identities and half-space Yang--Baxter random fields for stable spin Hall--Littlewood symmetric functions

本文证明双参数Littlewood恒等式，构造半空间Yang--Baxter随机场，给出采样规则并建立与六顶点模型的联系，获得Tracy--Widom极限。

One-dimensional quantum walks via generating function and the CGMV method

We treat a quantum walk (QW) on the line whose quantum coin at each vertex tends to be the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a "power-law" decay around the origin and a "strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at $x=0$ (the origin) and $x=1$ (the bottom) without continuous parts.

长上下文变换器的缩放极限

Scaling Limits of Long-Context Transformers

研究softmax自注意力在长上下文极限下的行为，分析逆温度参数对注意力选择性的影响，揭示不同尺度下注意力权重和输出的极限分布。