Edouard Brézin
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Edouard Brézin has a total of 6 co-authors. Here are the top co-authors by number of shared publications:
- Shinobu Hikami
Department of Basic Sciences University of Tokyo Meguro-ku Komaba 3-8-1, Tokyo 153 Japan
7 shared publications - Vladimir Kazakov
Laboratoire de Physique Théorique de l'Ecole Normale Supérieure et l' Université Paris-VI 24 rue Lhomond 75231 Paris CEDEX France
2 shared publications Ayumi Fujita
Department of Pure and Applied Sciences, University of Tokyo, Megroku, Komaba 3-8-1, Tokyo 153
1 shared publication- Cirano De Dominicis
Service de Physique Théorique CEA F-91191 Gif-sur-Yvette Cedex Saclay FRANCE
1 shared publication - Jean Zinn-Justin
Service de Physique Théorique † CE-Saclay 91191 Gif-sur-Yvette Cedex France
1 shared publication Shinobu Hikami
Department of Pure and Applied Sciences University of Tokyo Meguro-ku, Komaba 3-8-1 Tokyo 153, e
1 shared publication
Latest Publications
- Open Access
Characteristic polynomials of real symmetric random matrices
It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an exact dual representation of the problem: the $k$-point functions are expressed in terms of finite integrals over (quaternionic) $k\times k$ matrices. However the control of the Dyson limit, in which the distance of the various parameters $\la$'s is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson-Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a {\it finite} number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large $N$ limit.Open AccessCharacteristic polynomials of real symmetric random matrices
It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an exact dual representation of the problem: the $k$-point functions are expressed in terms of finite integrals over (quaternionic) $k\times k$ matrices. However the control of the Dyson limit, in which the distance of the various parameters $\la$'s is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson-Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a {\it finite} number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large $N$ limit. - 1 October 2000
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 214 (1) , 233-247
Universality of Correlations of Levels¶with Discrete Statistics
- 1 October 2000
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 214 (1) , 233-247
Universality of Correlations of Levels¶with Discrete Statistics
- 1 October 2000
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 214 (1) , 111-135
Characteristic Polynomials of Random Matrices
- 1 October 2000
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 214 (1) , 111-135
Characteristic Polynomials of Random Matrices
- 1 September 2000
- Published by American Physical Society (APS) in Physical Review E
- Vol. 62 (3) , 3558-3567
Characteristic polynomials of random matrices at edge singularities
We have discussed earlier the correlation functions of the random variables in which X is a random matrix. In particular, the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the instead of belonging to the bulk of the spectrum, approach the edge, a crossover takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of X, various different phenomenona may occur and one can tune the spectrum of this source matrix to other critical points. Again there are remarkably simple formulas for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.- 1 September 2000
- Published by American Physical Society (APS) in Physical Review E
- Vol. 62 (3) , 3558-3567
Characteristic polynomials of random matrices at edge singularities
We have discussed earlier the correlation functions of the random variables in which X is a random matrix. In particular, the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the instead of belonging to the bulk of the spectrum, approach the edge, a crossover takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of X, various different phenomenona may occur and one can tune the spectrum of this source matrix to other critical points. Again there are remarkably simple formulas for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.- Open Access
Interactions of several replicas in the random field Ising model
The replicated field theory of the random field Ising model involves the couplings of replicas of different indices. The resulting correlation functions involve a superposition of different types of long distance behaviours. However the $n=0$ limit allows one to discuss the renormalization group properties in spite of this phenomenon. The attraction of pairs of replicas is enhanced under renormalization flow and no stable fixed point is found. Consequently an instability occurs in the paramagnetic region, before one reaches the Curie line, signalling the onset of replica symmetry breaking.Open AccessInteractions of several replicas in the random field Ising model
The replicated field theory of the random field Ising model involves the couplings of replicas of different indices. The resulting correlation functions involve a superposition of different types of long distance behaviours. However the $n=0$ limit allows one to discuss the renormalization group properties in spite of this phenomenon. The attraction of pairs of replicas is enhanced under renormalization flow and no stable fixed point is found. Consequently an instability occurs in the paramagnetic region, before one reaches the Curie line, signalling the onset of replica symmetry breaking. - Open Access
Characteristic polynomials of random matrices at edge singularities
We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the $\la$'s, instead of belonging to the bulk of the spectrum, approach the edge, a cross-over takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of $X$, various new phenomenons may occur and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.Open AccessCharacteristic polynomials of random matrices at edge singularities
We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the $\la$'s, instead of belonging to the bulk of the spectrum, approach the edge, a cross-over takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of $X$, various new phenomenons may occur and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well. - Open Access
Logarithmic moments of characteristic polynomials of random matrices
In a recent article we have discussed the connections between averages of powers of Riemann's $\zeta$-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for $\zeta$-functions, and the correspondence is again striking.
Open AccessLogarithmic moments of characteristic polynomials of random matrices
In a recent article we have discussed the connections between averages of powers of Riemann's $\zeta$-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for $\zeta$-functions, and the correspondence is again striking.
- Open Access
Characteristic polynomials of random matrices
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power $K^2$ ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.Open AccessCharacteristic polynomials of random matrices
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power $K^2$ ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite. - Open Access
Universality of correlations of levels with discrete statistics
We study the statistics of a system of N random levels with integer values, in the presence of a logarithmic repulsive potential of Dyson type. This probleme arises in sums over representations (Young tableaux) of GL(N) in various matrix problems and in the study of statistics of partitions for the permutation group. The model is generalized to include an external source and its correlators are found in closed form for any N. We reproduce the density of levels in the large N and double scaling limits and the universal correlation functions in Dyson's short-distance scaling limit. We also study the statistics of small levels.
Open AccessUniversality of correlations of levels with discrete statistics
We study the statistics of a system of N random levels with integer values, in the presence of a logarithmic repulsive potential of Dyson type. This probleme arises in sums over representations (Young tableaux) of GL(N) in various matrix problems and in the study of statistics of partitions for the permutation group. The model is generalized to include an external source and its correlators are found in closed form for any N. We reproduce the density of levels in the large N and double scaling limits and the universal correlation functions in Dyson's short-distance scaling limit. We also study the statistics of small levels.
- Open Access
Perturbative analysis of an n-Ising model on a random surface
Two dimensional quantum gravity coupled to a conformally invariant matter field of central charge c=n/2, is represented, in a discretized version, by n independent Ising spins per cell of the triangulations of a random surface. The matrix integral representation of this model leads to a diagrammatic expansion at large orders, when the Ising coupling constant is tuned to criticality, one extracts the values of the string susceptibility exponent. We extend our previous calculation to order eight for genus zero and investigate now also the genus one case in order to check the possibility of having a well-defined double scaling limit even c>1.
Open AccessPerturbative analysis of an n-Ising model on a random surface
Two dimensional quantum gravity coupled to a conformally invariant matter field of central charge c=n/2, is represented, in a discretized version, by n independent Ising spins per cell of the triangulations of a random surface. The matrix integral representation of this model leads to a diagrammatic expansion at large orders, when the Ising coupling constant is tuned to criticality, one extracts the values of the string susceptibility exponent. We extend our previous calculation to order eight for genus zero and investigate now also the genus one case in order to check the possibility of having a well-defined double scaling limit even c>1.