The spectrum of the stochastic Bessel operator at high temperature

Abstract

Ramírez and Rider (2009) established that the hard edge of the spectrum of the $β$-Laguerre ensemble converges, in the high-dimensional limit, to the bottom of the spectrum of the stochastic Bessel operator. Using stochastic analysis tools, we prove that, in the high-temperature limit ($β\to 0$), the rescaled eigenvalue point process of this operator converges to a non-trivial limiting point process. This limit is characterized by a family of coupled diffusions and differs from a Poisson point process due to its interaction with the hard edge. Exploiting this diffusion characterization, we establish exact large deviation asymptotics for the largest eigenvalues. Furthermore, for an explicit range of the parameters, we relate this limiting process to the finite-$n$ $β$-Laguerre ensemble, conjecturing an exact distributional match with the infinite sum of its independent exponential gaps. As a byproduct of our analysis, we also formulate a conjecture regarding an explicit integral formula for the probability that a reflected Brownian motion with a constant drift hits an affine line, generalizing a formula of Salminen and Yor (2011).

Figures (2)

• Figure 1: Simulation of the diffusion $r_\mu$ for $\mu = 0.8$ and $a=1$.
• Figure 2: Simulation of two diffusions $r_\mu$ for $\mu = 0.5$ (blue) and $\mu =1$ (red), with $a=1$ (color online).

Theorems & Definitions (28)

• Theorem 1: Convergence of the eigenvalues
• Remark 1.1: Drifts versus critical line
• Theorem 2: Characterization of the limiting point process
• Theorem 3: First eigenvalue limit for $a \geq 0$
• Conjecture 1.2
• Proposition 1.3: Exact matching for $a=0$
• proof
• Conjecture 1.4: Matching for all $a \geq 0$
• Remark 1.5: Breakdown in the regime $a \in (-1, 0)$
• Proposition 1.6: Asymptotics for the largest $k$ eigenvalues