Large deviations for the $${\hbox {Sine}}_\beta $$ and $$\hbox {Sch}_\tau $$ processes

Overcrowding asymptotics for the $\operatorname{Sine}_{\beta}$ process

“…One of the consequences of this large deviation result is that if ρ > 1 2π then P (N β (λ) ≥ ρλ) decays asymptotically as e −βλ 2 I(ρ)(1+o(1)) as λ → ∞. It was shown in [10] that I(ρ) grows like 1 2 ρ 2 log ρ as ρ → ∞ so the decay of P (N β (λ) ≥ ρλ) is formally consistent with our Theorem 1, even though it describes a different regime of n, λ.…”

“…Outline of the proof. The estimates (13) are from the Appendix of [10]. The definition of W allows us to check that the inverse of the function 1 2 √ a log(16a) on [e 2 /16, ∞) is exactly x −2 W 2 (−x/4).…”

“…Under this transformationτ λ becomes the hitting time of ∞ for the process X λ . Following [10] we compare X λ with another family of diffusions. For a ≥ −1 consider the SDE…”

“…and denote by τ Y,λ,a the hitting time of ∞. In [10] the Cameron-Martin-Girsanov formula was used for diffusions with explosions to compare X λ and Y λ,a and the following results were obtained.…”

“…Our proof uses techniques that we developed in [10], where we proved a large deviation principle for 1 λ N β (λ) with scale λ 2 and a good rate function of the form βI(ρ). One of the consequences of this large deviation result is that if ρ > 1 2π then P (N β (λ) ≥ ρλ) decays asymptotically as e −βλ 2 I(ρ)(1+o(1)) as λ → ∞.…”