论文标题
在变形的皮尔西决定因素上
On the deformed Pearcey determinant
论文作者
论文摘要
在本文中，我们关注的是珍珠决定因素$ \ det \ left（i-γk^{\ mathrm {pe}} _ {s，ρ} \ right）$，其中$ 0 \ leq leqleqγ<1 $ and $ k^{\ kathrm {pe}} $ l^2 \ left（-s，s \右）$，带有随机矩阵理论引起的经典皮尔西内核。该决定因素对应于稀疏后皮尔西过程的间隙概率，这意味着皮尔西过程中的每个粒子都以$ 1-γ$的概率独立去除。我们建立了涉及与非线性微分方程系统的特殊解决方案家族相关的涉及哈密顿量的变形皮尔西决定因素的整体表示。加上对哈密顿量的一些显着的差分身份，这使我们能够获得较大的差距渐近学，包括恒定术语的确切计算，这补充了我们先前在未经证件的情况下的工作（即$γ= 1 $）。出来的是，变形的皮尔西决定因素与未呈现的情况表现出明显不同的渐近行为，这表明随着参数$γ$的变化，将发生过渡。作为结果的应用，我们获得了降级函数的期望和差异的渐近学，以及中心限制定理。
In this paper, we are concerned with the deformed Pearcey determinant $\det\left(I-γK^{\mathrm{Pe}}_{s,ρ}\right)$, where $0 \leq γ<1$ and $K^{\mathrm{Pe}}_{s,ρ}$ stands for the trace class operator acting on $L^2\left(-s, s\right)$ with the classical Pearcey kernel arising from random matrix theory. This determinant corresponds to the gap probability for the Pearcey process after thinning, which means each particle in the Pearcey process is removed independently with probability $1-γ$. We establish an integral representation of the deformed Pearcey determinant involving the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations. Together with some remarkable differential identities for the Hamiltonian, this allows us to obtain the large gap asymptotics, including the exact calculation of the constant term, which complements our previous work on the undeformed case (i.e., $γ=1$). It comes out that the deformed Pearcey determinant exhibits a significantly different asymptotic behavior from the undeformed case, which suggests a transition will occur as the parameter $γ$ varies. As an application of our results, we obtain the asymptotics for the expectation and variance of the counting function for the Pearcey process, and a central limit theorem as well.
