Gibbs measures asymptotics

Abstract

Let (Ω, B, ν) be a measure space and H: Ω → ℝ⁺ be B measurable. Let ∫_Ω e ^−H dν < ∞. For 0 < T < 1 let µ _H,T (·) be the probability measure defined by

$$ \mu _{H,T} (A) \equiv \left( {\int_A {e^{ - H/T} d\nu } } \right)/\left( {\int_\Omega {e^{ - H/T} d\nu } } \right),A \in \mathcal{B}. $$

In this paper, we study the behavior of µ _H,T (·) as T ↓ 0 and extend the results of Hwang (1980, 1981). When Ω is ℝ and H achieves its minimum at a single value x ₀ (single well case) and H(·) is Hölder continuous at x ₀ of order α, it is shown that if X _T is a random variable with probability distribution µ _H,T (·) then as T ↓ 0, i) X _T → x ₀ in probability; ii) (X _t − x ₀)T ^−1/α converges in distribution to an absolutely continuous symmetric distribution with density proportional to $e^{ - c_\alpha |x|^\alpha }$ for some 0 < c _α < ∞. This is extended to the case when H achieves its minimum at a finite number of points (multiple well case). An extension of these results to the case H: ℝⁿ → ℝ⁺ is also outlined.