Determinants of Laplacians on converging hyperbolic surfaces
Abstract
Let Sk be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of Sk converges to a constant depending only the law of the limiting surface.
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