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 - Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials
AU - Beliaev, Dmitry
AU - Muirhead, Stephen
AU - Wigman, Igor
PY - 2020/12/10
Y1 - 2020/12/10
N2 - Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations.The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a "typical" real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus.
AB - Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations.The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a "typical" real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus.
M3 - Article
JO - Annales de l’Institut Henri Poincaré, Probabilités et Statistiques
JF - Annales de l’Institut Henri Poincaré, Probabilités et Statistiques
SN - 0246-0203
ER -
TY - JOUR
T1 - Mean conservation of nodal volume and connectivity measures for Gaussian ensembles
AU - Beliaev, Dmitry
AU - Muirhead, Stephen
AU - Wigman, Igor
PY - 2020/11/20
Y1 - 2020/11/20
M3 - Article
JO - ADVANCES IN MATHEMATICS
JF - ADVANCES IN MATHEMATICS
SN - 0001-8708
ER -