Spectral gap of Boltzmann measures on unit circle (Q2048177)

Let \(\mu_h\) be a Boltzmann measure on the unit circle in \(\mathbb{R}^2\) with parameter \(h\in\mathbb{R}\), and \(\lambda_1(\mu_h)\) be its spectral gap. The authors prove an explicit two-sided bound on this spectral gap: \[ 1\vee\frac{|h|}{7}\leq\lambda_1(\mu_h)\leq\sqrt{3}|h|+\frac{2h^2+3}{h^2+3}\,. \] This bound is sharp at \(h=0\) and gives a rate of convergence of \(\lambda_1(\mu_h)\) to infinity as \(h\to\infty\).