scholarly journals A Renormalisation Group Approach to the Universality of Wigner’s Semicircle Law for Random Matrices with Dependent Entries

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Thomas Krajewski
Keyword(s):  
Renormalisation Group ◽  
Group Equation ◽  
Replica Technique ◽  
Group Approach ◽  
Semicircle Law ◽  
Random Matrix Model ◽  
The Matrix ◽  
Renormalisation Group Equation ◽  
Non Gaussian ◽  
Dependent Entries

We show that if the non-Gaussian part of the cumulants of a random matrix model obeys some scaling bounds in the size of the matrix, then Wigner’s semicircle law holds. This result is derived using the replica technique and an analogue of the renormalisation group equation for the replica effective action.

scholarly journals A renormalisation group equation for transport coefficients in (2 + 1)-dimensions derived from the AdS/CMT correspondence

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Brian P. Dolan
Keyword(s):  
Magnetic Field ◽  
Charge Density ◽  
Transport Coefficients ◽  
Space Time ◽  
Two Dimensions ◽  
Renormalisation Group ◽  
Ac Conductivities ◽  
Group Equation ◽  
De Sitter ◽  
Renormalisation Group Equation

Abstract Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in four space-time dimensions can be used to analyse transport properties in two space dimensions. A non-linear renormalisation group equation for the conductivity in two dimensions is derived in this model and, as an example of its application, both the Ohmic and Hall DC and AC conductivities are studied in the presence of a magnetic field, using a bulk dyonic solution of the Einstein-Maxwell equations in asymptotically AdS4 space-time. The $$ \mathcal{Q} $$ Q -factor of the cyclotron resonance is shown to decrease as the temperature is increased and increase as the charge density is increased in a fixed magnetic field. Likewise the dissipative Ohmic conductivity at resonance increases as the temperature is decreased and as the charge density is increased. The analysis also involves a discussion of the piezoelectric effect in the context of the AdS/CMT framework.

LOW TEMPERATURE PROPERTIES OF 2D CORRELATED ELECTRONS IN WEAKLY DISORDERED MATERIALS

2003 ◽  
Vol 17 (28) ◽  
pp. 4987-4997
Author(s):  
DAVID NEILSON ◽  
D. J. W. GELDART
Keyword(s):  
Phase Transition ◽  
Temperature Dependence ◽  
Quantum Phase Transition ◽  
Direct Evidence ◽  
Coulomb Repulsion ◽  
Electron System ◽  
Renormalisation Group ◽  
Group Equation ◽  
Renormalisation Group Equation

Transport properties of extremely high purity two-dimensional (2D) electron systems at low temperatures are still not well understood either experimentally or theoretically, even though these systems are fast becoming a mainstream basis of computing devices. In fact there are two separate issues to be resolved. The first of these has attracted the more attention. This is the existence of a quantum phase transition (the metal-insulator transition) in the low density 2D system at zero temperature. Experimentally, in spite of claims, from existing data at finite temperatures there is no conclusive evidence either way on the existence of a T = 0 quantum phase transition. There is a need for a unified theory encompassing, on the same level, both insulating and metallic behaviour to predict the cross-over. We propose a semi-empirical one parameter renormalisation group equation for the temperature dependent resistivity of a 2D electron system with weak disorder. The renormalisation group equation has a physically meaningful insulating limit and it predicts a metallic ground state of zero resistance at higher electron densities. The resulting temperature dependence of the resistivity is found to give a good fit to experimental data near the separatrix. The second issue is the mechanism behind the sudden change in the temperature dependence of the resistivity, as is actually observed at low but non-zero temperatures, T = 0.1 to 2 K. This phenomenon is well-documented experimentally and it is of interest in its own right whether or not there is an actual transition at T = 0. We present direct evidence of the important role of the electron Coulomb repulsion and exchange in determining these finite temperature properties by noting an empirical relationship between the critical density at the bifurcation point and parallel magnetic field. The relationship is controlled by properties of the electron-electron correlation function for the 2D electron system. This result provides direct evidence of the central role of the Coulomb repulsion and exchange in driving the bifurcation phenomenon.

Identifying graph clusters using variational inference and links to covariance parametrization

2009 ◽  
Vol 367 (1906) ◽  
pp. 4407-4426
Author(s):  
David Barber
Keyword(s):  
Mean Field ◽  
Matrix Decomposition ◽  
Difficult Problem ◽  
Positive Definite ◽  
Variational Inference ◽  
Field Theories ◽  
Variational Approximation ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Non Gaussian

Finding clusters of well-connected nodes in a graph is a problem common to many domains, including social networks, the Internet and bioinformatics. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We use a clique matrix decomposition based on a statistical description that encourages clusters to be well connected and few in number. The formal intractability of inferring the clusters is addressed using a variational approximation inspired by mean-field theories in statistical mechanics. Clique matrices also play a natural role in parametrizing positive definite matrices under zero constraints on elements of the matrix. We show that clique matrices can parametrize all positive definite matrices restricted according to a decomposable graph and form a structured factor analysis approximation in the non-decomposable case. Extensions to conjugate Bayesian covariance priors and more general non-Gaussian independence models are briefly discussed.

scholarly journals The almost sure semicircle law for random band matrices with dependent entries

2021 ◽  
Vol 131 ◽  
pp. 172-200
Author(s):  
Michael Fleermann ◽  
Werner Kirsch ◽  
Thomas Kriecherbauer
Keyword(s):  
Band Matrices ◽  
Semicircle Law ◽  
Random Band ◽  
Dependent Entries
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