TY - JOUR
T1 - The band structure of a model of spatial random permutation
AU - Fyodorov, Yan
AU - Muirhead, Stephen
N1 - Funding Information:
This research was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grant EP/N009436/1 “The many faces of random characteristic polynomials” and the Australian Research Council (ARC) Discovery Early Career Researcher Award DE200101467. The authors would like to thank Jeremiah Buckley, Naomi Feldheim and Daniel Ueltschi for enlightening discussions, and in particular Ron Peled for helpful discussions at an early stage. The authors would also like to thank an anonymous referee for detailed comments which improved the presentation of the paper, and also for pointing out corrections to an earlier version.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/2/7
Y1 - 2021/2/7
N2 - We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size N tends to infinity and the inverse temperature β tends to zero; in particular, we show that the mean displacement is of order min { 1 / β, N}. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac–Murdock–Szegő matrices.
AB - We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size N tends to infinity and the inverse temperature β tends to zero; in particular, we show that the mean displacement is of order min { 1 / β, N}. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac–Murdock–Szegő matrices.
UR - http://www.scopus.com/inward/record.url?scp=85100568853&partnerID=8YFLogxK
U2 - 10.1007/s00440-020-01019-z
DO - 10.1007/s00440-020-01019-z
M3 - Article
VL - 179
SP - 543
EP - 587
JO - PROBABILITY THEORY AND RELATED FIELDS
JF - PROBABILITY THEORY AND RELATED FIELDS
SN - 0178-8051
IS - 3-4
ER -
TY - JOUR
T1 - Nonlinearity-generated resilience in large complex systems
AU - Belga Fedeli, Sirio
AU - Fyodorov, Yan
AU - Ipsen, J R
N1 - Funding Information:
We acknowledge support by the Engineering and Physical Sciences Research Council Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems through Grant No. EP/L015854/1 (S.B.F.) and by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (J.R.I.).
Publisher Copyright:
© 2021 American Physical Society.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
PY - 2021/2/1
Y1 - 2021/2/1
N2 - We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a “resilience gap”: there are no other fixed points within a radius r∗>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r∗. The radius r∗ is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.
AB - We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a “resilience gap”: there are no other fixed points within a radius r∗>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r∗. The radius r∗ is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.
UR - http://www.scopus.com/inward/record.url?scp=85101286743&partnerID=8YFLogxK
U2 - 10.1103/PhysRevE.103.022201
DO - 10.1103/PhysRevE.103.022201
M3 - Article
VL - 103
JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
IS - 2
M1 - 022201
ER -