TY - JOUR

T1 - Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices

AU - Valov, Alexander

AU - Meerson, Baruch

AU - Sasorov, Pavel V.

N1 - Publisher Copyright:
© 2024 The Author(s). Published by IOP Publishing Ltd.

PY - 2024/2/9

Y1 - 2024/2/9

N2 - We evaluate, in the large-N limit, the complete probability distribution P ( A , m ) of the values A of the sum ∑ i = 1 N | λ i | m , where λ i ( i = 1 , 2 , … , N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang-Landau algorithm, we found that, in the limit of N → ∞ , the rate function of P ( A , m ) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue ‘evaporates’ from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p = 1 + 1 / | m − 1 | , where 0 < m < 2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m ⩽ 1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner’s semicircular distribution.

AB - We evaluate, in the large-N limit, the complete probability distribution P ( A , m ) of the values A of the sum ∑ i = 1 N | λ i | m , where λ i ( i = 1 , 2 , … , N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang-Landau algorithm, we found that, in the limit of N → ∞ , the rate function of P ( A , m ) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue ‘evaporates’ from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p = 1 + 1 / | m − 1 | , where 0 < m < 2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m ⩽ 1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner’s semicircular distribution.

KW - Coulomb gas

KW - Gaussian random matrices

KW - Wang-Landau algorithm

KW - large deviations

KW - phase transitions

KW - spectral linear statistics

UR - http://www.scopus.com/inward/record.url?scp=85184029025&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/ad1e1a

DO - 10.1088/1751-8121/ad1e1a

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AN - SCOPUS:85184029025

SN - 1751-8113

VL - 57

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 6

M1 - 065001

ER -