Replica saddle-point approximation and analytic continuation

Establish a detailed analysis of the saddle-point approximation used to reduce the replica permutation sum to the minimum generalized entropy over cuts, including whether analytic continuation from integer replica number commutes with minimization when multiple cuts contribute.

Background

The random-matter calculation produces a sum over cuts weighted by Rényi entropies. The paper assumes replica symmetry and a sufficiently large entropy gap so that one cut dominates the sum for integer replica number greater than one.

The authors note that the suppression of nonminimal cuts degenerates as the replica index approaches one. If analytic continuation does not commute with minimization, several cuts can contribute and the sharp minimum is replaced by a smoothed result. The paper does not analyze this issue in detail.