TY - JOUR
T1 - Extreme values of CUE characteristic polynomials
T2 - a numerical study
AU - Fyodorov, Yan V.
AU - Gnutzmann, Sven
AU - Keating, Jonathan P.
PY - 2018/10/22
Y1 - 2018/10/22
N2 - We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.
AB - We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.
KW - freezing transition
KW - log-correlated processes
KW - random matrix theory
UR - http://www.scopus.com/inward/record.url?scp=85055470867&partnerID=8YFLogxK
U2 - 10.1088/1751-8121/aae65a
DO - 10.1088/1751-8121/aae65a
M3 - Article
AN - SCOPUS:85055470867
VL - 51
JO - Journal of Physics A
JF - Journal of Physics A
SN - 1751-8113
IS - 46
M1 - 464001
ER -
TY - BOOK
T1 - Stochastic processes and random matrices
T2 - Lecture notes of the Les Houches summer school
AU - Schehr, Grégory
AU - Altland, Alexander
AU - Fyodorov, Yan V.
AU - O'Connell, Neil
AU - Cugliandolo, Leticia F.
PY - 2018/1/18
Y1 - 2018/1/18
N2 - The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).
AB - The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).
UR - http://www.scopus.com/inward/record.url?scp=85052640972&partnerID=8YFLogxK
U2 - 10.1093/oso/9780198797319
DO - 10.1093/oso/9780198797319
M3 - Book
AN - SCOPUS:85052640972
SN - 9780198797319
VL - 104
BT - Stochastic processes and random matrices
PB - Oxford University Press
ER -