TY - JOUR

T1 - Universal scaling of the tail of the density of eigenvalues in random matrix models

AU - Bowick, Mark J.

AU - Brézin, Edouard

N1 - Funding Information:
This researchw as supportedb y the Outstanding Junior Investigator Grant DOE DE-FG02-85ER40231a ndby NSF grantP HY 89-04035B. oth authorsw ouldl iketo thankt heI nstitutefo r Theoretical Physicsa ndits stafff or providingth estimulating environmenwt hich made this work possible.E . Brrzin thanksS . Shenkefro r a stimulatindgi scussion.

PY - 1991/10/3

Y1 - 1991/10/3

N2 - Large random matrices have eigenvalue density distributions limited to a finite support. Near the endpoint of the support, when the size N of the matrices is large, one can study a scaling region of size N-μ in which the cross-over from a non-zero density to a vanishing density takes place. This cross-over is shown to be universal (for random hermitian matrices with a unitary-invariant probability distribution), in the sense that it depends only on the order of multicriticality of the problem. For a multicritical point of odd order k, the large-N density vanishes as |λ-λc|k- 1 2 near λc. The cross-over function of the scaling variables (λc-λ)Nμ and (gc-g)Nν (where g is the coupling constant characterizing the potential or equivalently the cosmological constant of 2D quantum gravity) is related to the resolvent of the Schrödinger operator in which the potential is the scaling function which satisfies the string equation. The exponents are found to be μ= 2 (2k+1) and ν= 2k (2k+1).

AB - Large random matrices have eigenvalue density distributions limited to a finite support. Near the endpoint of the support, when the size N of the matrices is large, one can study a scaling region of size N-μ in which the cross-over from a non-zero density to a vanishing density takes place. This cross-over is shown to be universal (for random hermitian matrices with a unitary-invariant probability distribution), in the sense that it depends only on the order of multicriticality of the problem. For a multicritical point of odd order k, the large-N density vanishes as |λ-λc|k- 1 2 near λc. The cross-over function of the scaling variables (λc-λ)Nμ and (gc-g)Nν (where g is the coupling constant characterizing the potential or equivalently the cosmological constant of 2D quantum gravity) is related to the resolvent of the Schrödinger operator in which the potential is the scaling function which satisfies the string equation. The exponents are found to be μ= 2 (2k+1) and ν= 2k (2k+1).

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U2 - 10.1016/0370-2693(91)90916-E

DO - 10.1016/0370-2693(91)90916-E

M3 - Article

AN - SCOPUS:0002704693

SN - 0370-2693

VL - 268

SP - 21

EP - 28

JO - Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

JF - Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

IS - 1

ER -