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Glass-like transition described by toppling of stability hierarchy* *Dedicated to the memory of Fritz Haake.

Authors
  • Grela, Jacek1
  • Khoruzhenko, Boris A2
  • 1 Jagiellonian University, Poland , (Poland)
  • 2 Queen Mary University of London, United Kingdom , (United Kingdom)
Type
Published Article
Journal
Journal of Physics A: Mathematical and Theoretical
Publisher
IOP Publishing
Publication Date
Mar 15, 2022
Volume
55
Issue
15
Identifiers
DOI: 10.1088/1751-8121/ac56aa
Source
ioppublishing
Keywords
Disciplines
  • Advances in Quantum Chaos, Random-Matrix Theory and the Semiclassical Limit: In Memory of Fritz Haak
License
Unknown

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.

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