Glass-like transition described by toppling of stability hierarchy*

Abstract

Abstract Building on the work of Fyodorov (2004) and Fyodorov and Nadal (2012) we examine the critical behaviour of population of saddles with fixed instability index k in high dimensional random energy landscapes. Such landscapes consist of a parabolic confining potential and a random part in N ≫ 1 dimensions. When the relative strength m of the parabolic part is decreasing below a critical value m c, the random energy landscapes exhibit a glass-like transition from a simple phase with very few critical points to a complex phase with the energy surface having exponentially many critical points. We obtain the annealed probability distribution of the instability index k by working out the mean size of the population of saddles with index k relative to the mean size of the entire population of critical points and observe toppling of stability hierarchy which accompanies the underlying glass-like transition. In the transition region m = m c + δN −1/2 the typical instability index scales as k = κN 1/4 and the toppling mechanism affects whole instability index distribution, in particular the most probable value of κ changes from κ = 0 in the simple phase (δ > 0) to a non-zero value κ max ∝ (−δ)3/2 in the complex phase (δ < 0). We also show that a similar phenomenon is observed in random landscapes with an additional fixed energy constraint and in the p-spin spherical model.