arXiv 1306.41062013-06math-phmath.MP

实Ginibre系综的扩散过程和渐近体能隙几率

Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble

Peter J. Forrester

备注：: 15 pages; version two corrections the reported value of c_2, as alerted to me by J. Edge
发表信息：: J. Phys. A: Math. Theor. 48 324001 (2015)

摘要：

已知的是，真实Ginibre系综的实本征值的体积标度极限在分布上等于湮灭过程$A + A \to \emptyset$的重新标度的$t \to \infty$极限。此外，在聚结过程$A + A \to A$的重新标度的$t \to \infty$极限中随机地删除每个粒子，产生在分布上与湮灭过程相等的过程。我们使用这些相互关系推导出从现有的文献中的渐近小和大的距离形式的间隙概率为真正的Ginibre合奏。特别地，后者的前导形式被示出为等于$\exp(-(ζ(3/2)/(2 \sqrt{2 π}))s)$，其中$s$表示间隙大小，并且$ζ(z)$表示黎曼ζ函数。它示出了如何可以严格地建立使用矩阵Fredholm算子的渐近公式。在有限的$N$情况下，间隙概率的行列式公式推导，这是用来说明对数值计算的渐近公式。

英文摘要：

It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled $t \to \infty$ limit of the annihilation process $A + A \to \emptyset$. Furthermore, deleting each particle at random in the rescaled $t \to \infty$ limit of the coalescence process $A + A \to A$, a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to $\exp(-(ζ(3/2)/(2 \sqrt{2 π}))s)$, where $s$ denotes the gap size and $ζ(z)$ denotes the Riemann zeta function. It is shown how this can be rigorously established using an asymptotic formula for matrix Fredholm operators. A determinant formula is derived for the gap probability in the finite $N$ case, and this is used to illustrate the asymptotic formulas against numerical computations.

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摘要：

英文摘要：

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