# A singular-potential random matrix model arising in mean-field glassy systems

Akemann G, Villamaina D, Vivo P (2014) *Physical Review E* 89(6).

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We consider an invariant random matrix model where the standard Gaussianpotential is distorted by an additional single pole of order $m$. We computethe average or macroscopic spectral density in the limit of large matrix size,solving the loop equation with the additional constraint of vanishing trace onaverage. The density is generally supported on two disconnected intervals lyingon the two sides of the pole. In the limit of having no pole, we recover thestandard semicircle. Obtained in the planar limit, our results apply tomatrices with orthogonal, unitary or symplectic symmetry alike. The orthogonalcase with $m=2$ is motivated by an application to spin glass physics. In theSherrington-Kirkpatrick mean-field model, in the paramagnetic phase and forsufficiently large systems the spin glass susceptibility is a random variable,depending on the realization of disorder. It is essentially given by a linearstatistics on the eigenvalues of the coupling matrix. As such its largedeviation function can be computed using standard Coulomb fluid techniques. Theresulting free energy of the associated fluid precisely corresponds to thepartition function of our random matrix model. Numerical simulations provide anexcellent confirmation of our analytical results.

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Akemann G, Villamaina D, Vivo P. A singular-potential random matrix model arising in mean-field glassy systems.

*Physical Review E*. 2014;89(6).Akemann, G., Villamaina, D., & Vivo, P. (2014). A singular-potential random matrix model arising in mean-field glassy systems.

*Physical Review E*,*89*(6). doi:10.1103/physreve.89.062146Akemann, G., Villamaina, D., and Vivo, P. (2014). A singular-potential random matrix model arising in mean-field glassy systems.

*Physical Review E*89.Akemann, G., Villamaina, D., & Vivo, P., 2014. A singular-potential random matrix model arising in mean-field glassy systems.

*Physical Review E*, 89(6). G. Akemann, D. Villamaina, and P. Vivo, “A singular-potential random matrix model arising in mean-field glassy systems”,

*Physical Review E*, vol. 89, 2014. Akemann, G., Villamaina, D., Vivo, P.: A singular-potential random matrix model arising in mean-field glassy systems. Physical Review E. 89, (2014).

Akemann, Gernot, Villamaina, Dario, and Vivo, Pierpaolo. “A singular-potential random matrix model arising in mean-field glassy systems”.

*Physical Review E*89.6 (2014).
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PMID: 25019763

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arXiv 1402.0755

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