论文标题
汉克尔的决定因素和正交多项式，用于扰动高斯的重量：从有限$ n $到大$ n $渐近。
Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
论文作者
论文摘要
我们在对称扰动的高斯重量$$ w（x; t）：= \ mathrm {e}^{ - x^2} \ left（1+t \：x^2 \ right）^λ，\ qquad x x \ in \ qquad in \ quthbb = r} in p（1+t）中，研究了对称扰动的高斯重量$$ W（x; t）：= \ MATHRM {E}^{ - x^2} \ qquad x \ in \ mathbb = \ mathbb {r} $。该权重与信息理论中的单用户MIMO系统有关。结果表明，复发系数$β_n（t）$与特定的painlevév vryscentent有关，子领导系数$ \ mathrm {p}（n，t）$满足jimbo-miwa-okamoto $σ$ - pachlevévEquination的pachlevéVequination。此外，我们分别通过$β_N（t）$和$ \ mathrm {p}（n，t）$得出二阶差方程。这使我们能够借助Dyson的库仑液体方法获得$β_N（t）$和$ \ MATHRM {p}（n，t）$的大$ n $全渐近扩展。我们还考虑由扰动的高斯重量产生的hankel行列式$ d_n（t）$。发现$ h_n（t）$，与$ d_n（t）$的对数衍生物相关的数量，可以用$β_N（t）$和$ \ m atrm {p}（n，t）$表示。基于此结果，我们获得了$ h_n（t）$的大$ n $渐近扩展，然后获得hankel decentant $ d_n（t）$的大型扩展。
We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\: x^2\right)^λ,\qquad x\in \mathbb{R}, $$ where $t> 0,\;λ\in \mathbb{R}$. This weight is related to the single-user MIMO systems in information theory. It is shown that the recurrence coefficient $β_n(t)$ is related to a particular Painlevé V transcendent, and the sub-leading coefficient $\mathrm{p}(n,t)$ satisfies the Jimbo-Miwa-Okamoto $σ$-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by $β_n(t)$ and $\mathrm{p}(n,t)$, respectively. This enables us to obtain the large $n$ full asymptotic expansions for $β_n(t)$ and $\mathrm{p}(n,t)$ with the aid of Dyson's Coulomb fluid approach. We also consider the Hankel determinant $D_n(t)$, generated by the perturbed Gaussian weight. It is found that $H_n(t)$, a quantity allied to the logarithmic derivative of $D_n(t)$, can be expressed in terms of $β_n(t)$ and $\mathrm{p}(n,t)$. Based on this result, we obtain the large $n$ asymptotic expansion of $H_n(t)$ and then that of the Hankel determinant $D_n(t)$.
