Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble
Abstract
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 interrelationships 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 $\exp((\zeta(3/2)/(2 \sqrt{2 \pi}))s)$, where $s$ denotes the gap size and $\zeta(z)$ 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2013
 arXiv:
 arXiv:1306.4106
 Bibcode:
 2013arXiv1306.4106F
 Keywords:

 Mathematical Physics
 EPrint:
 15 pages