TY - JOUR
T1 - Towards rigorous analysis of the Levitov-Mirlin-Evers recursion
AU - Fyodorov, Yan V
AU - Kupiainen, Antti
AU - Webb, Christian
PY - 2016/11/10
Y1 - 2016/11/10
N2 - This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities of LME recursion to those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent q. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study
AB - This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities of LME recursion to those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent q. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study
U2 - 10.1088/0951-7715/29/12/3871
DO - 10.1088/0951-7715/29/12/3871
M3 - Article
VL - 29
JO - NONLINEARITY
JF - NONLINEARITY
SN - 0951-7715
IS - 12
M1 - 3871
ER -
TY - CHAP
T1 - Random Matrix Theory of resonances
T2 - 2016 URSI International Symposium on Electromagnetic Theory, EMTS 2016
AU - Fyodorov, Yan V.
PY - 2016/9/19
Y1 - 2016/9/19
N2 - Scattering of electromagnetic waves in billiard-like systems has become a standard experimental tool of studying properties associated with Quantum Chaos. Random Matrix Theory (RMT) describing statistics of eigenfrequencies and associated eigenfunctions remains one of the pillars of theoretical understanding of quantum chaotic systems. In a scattering system coupling to continuum via antennae converts real eigenfrequencies into poles of the scattering matrix in the complex frequency plane and the associated eigenfunctions into decaying resonance states. Understanding statistics of these poles, as well as associated non-orthogonal resonance eigenfunctions within RMT approach is still possible, though much more challenging task.
AB - Scattering of electromagnetic waves in billiard-like systems has become a standard experimental tool of studying properties associated with Quantum Chaos. Random Matrix Theory (RMT) describing statistics of eigenfrequencies and associated eigenfunctions remains one of the pillars of theoretical understanding of quantum chaotic systems. In a scattering system coupling to continuum via antennae converts real eigenfrequencies into poles of the scattering matrix in the complex frequency plane and the associated eigenfunctions into decaying resonance states. Understanding statistics of these poles, as well as associated non-orthogonal resonance eigenfunctions within RMT approach is still possible, though much more challenging task.
UR - http://www.scopus.com/inward/record.url?scp=84992128522&partnerID=8YFLogxK
U2 - 10.1109/URSI-EMTS.2016.7571486
DO - 10.1109/URSI-EMTS.2016.7571486
M3 - Other chapter contribution
AN - SCOPUS:84992128522
SP - 666
EP - 669
BT - 2016 URSI International Symposium on Electromagnetic Theory, EMTS 2016
PB - Institute of Electrical and Electronics Engineers Inc.
Y2 - 14 August 2016 through 18 August 2016
ER -