The Gaussian free field in an interlacing particle system with two jump rates

带跳跃的高斯权正交多项式与离散潘勒韦方程

Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations

本文提出精细化的离散潘勒韦识别问题，强调超越表面类型描述动力学，并以带单跳跃高斯权正交多项式中的递推关系为例展示求解方法。

带跳跃间断点的高斯酉系综、偏微分方程与耦合Painlevé IV系统

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System

该研究针对带跳跃间断点的高斯权重生成的Hankel行列式，借助阶梯算子与补充条件推导其对数导数满足的二阶偏微分方程，进一步通过Riemann-Hilbert问题建立辅助量与耦合Painlevé IV系统解的关联。

Quantum-jump vs stochastic Schrödinger dynamics for Gaussian states with quadratic Hamiltonians and linear Lindbladians

The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian s…

The Vertex Reinforced Jump Process and a Random Schrödinger operator on finite graphs

We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schrödinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised b…

Dynkin isomorphism and Mermin--Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin--Wagner theorem for these sigma models. This result is of intrinsic interest for the s…

随机哈密顿量生成的幺正演化的电路复杂度的极值跳跃

Extremal jumps of circuit complexity of unitary evolutions generated by random Hamiltonians

本文研究了随机哈密顿量生成的幺正演化的电路复杂度，发现其以高概率在逃离恒等邻域的时间尺度内即达到最大值，并利用测度集中技术实现了更精细的控制。

Spin-fluctuation theory beyond Gaussian approximation

A characteristic feature of the Gaussian approximation in the functional-integral approach to the spin-fluctuation theory is the jump phase transition to the paramagnetic state. We eliminate the jump and obtain a continuous second-order phase transition by taking into account high-order terms in the expansion of the free energy in powers of the fluctuating exchange field. The third-order term of the free energy renormalizes the mean field, and fourth-order term, responsible for the interaction of the fluctuations, renormalizes the spin susceptibility. The extended theory is applied to the calc…

Statistics of the Number of Records for Random Walks and Lévy Flights on a ${1D}$ Lattice

We study the statistics of the number of records $R_n$ for a symmetric, $n$-step, discrete jump process on a $1D$ lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution $P(R_n)$ of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival pro…

Computing the Rates of Measurement-Induced Quantum Jumps

Small quantum systems can now be continuously monitored experimentally which allows for the reconstruction of quantum trajectories. A peculiar feature of these trajectories is the emergence of jumps between the eigenstates of the observable which is measured. Using the Stochastic Master Equation (SME) formalism for continuous quantum measurements, we show that the density matrix of a system indeed shows a jumpy behavior when it is subjected to a tight measurement (even if the noise in the SME is Gaussian). We are able to compute the jump rates analytically for any system evolution, i.e. any Li…

高斯态的GKLS向量场动力学

GKLS Vector Field Dynamics for Gaussian States

本文为高斯态系统构造了GKLS生成元对应的向量场，证明其可分解为哈密顿、类梯度和跳跃三部分，并用谐振子耗散实例加以说明。