1110.4284 (Peter J. Forrester et al.)

Asymptotic forms for hard and soft edge general $β$ conditional gap
probabilities    [PDF]

Peter J. Forrester, Nicholas S. Witte

An infinite log-gas formalism, due to Dyson, and independently Fogler and
Shklovskii, is applied to the computation of conditioned gap probabilities at
the hard and soft edges of random matrix $\beta$-ensembles. The conditioning is
that there are $n$ eigenvalues in the gap, with $n \ll |t|$, $t$ denoting the
end point of the gap. It is found that the entropy term in the formalism must
be replaced by a term involving the potential drop to obtain results consistent
with known asymptotic expansions in the case $n=0$. With this modification made
for general $n$, the derived expansions - which are for the logarithm of the
gap probabilities - are conjectured to be correct up to and including terms
O$(\log|t|)$. They are shown to satisfy various consistency conditions,
including an asymptotic duality formula relating $\beta$ to $4/\beta$.

View original: http://arxiv.org/abs/1110.4284