Glass-like transition described by toppling of stability hierarchy*

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