Related papers: A smooth transition from Wishart to GOE
We consider high-dimensional Wishart matrices $d^{-1}\mathcal{X}_{n,d}\mathcal{X}_{n,d}^T$, associated with a rectangular random matrix $\mathcal{X}_{n,d}$ of size $n\times d$ whose entries are jointly Gaussian and correlated. Even if we…
We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nested submatrices one obtains a limiting…
We consider the fractional $\Phi^3_d$-measure on the $d$-dimensional torus, with Gaussian free field having inverse covariance $(1-\Delta)^\alpha$, and show a phase transition at $d=3\alpha$. More precisely, in a regular regime $d<3\alpha$,…
Until now only for specific crossovers between Poissonian statistics (P), the statistics of a Gaussian orthogonal ensemble (GOE), or the statistics of a Gaussian unitary ensemble (GUE) analytical formulas for the level spacing distribution…
We study the spectral properties of and spectral-crossovers between different random matrix ensembles (Poissonian, GOE, GUE) in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term,…
Consider a high-dimensional Wishart matrix $\bd{W}=\bd{X}^T\bd{X}$ where the entries of $\bd{X}$ are i.i.d. random variables with mean zero, variance one, and a finite fourth moment $\eta$. Motivated by problems in signal processing and…
We analyze in detail a second order phase transition that occurs in large N Gaussian multi-matrix models in which the matrices are constrained to be commuting. The phase transition occurs as the relative masses of the matrices are varied,…
Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian…
We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The…
We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described…
The large N_c N=4 gauge theory with quenched N=2 quark matter in the presence of a magnetic field displays chiral symmetry breaking. We study the temperature and chemical potential dependence of this theory using its gravity dual (based on…
We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt…
The order parameter of a quantum-coherent many-body system can include a phase degree of freedom, which, in the presence of an electromagnetic field, depends on the choice of gauge. Because of the relationship between the phase gradient and…
The deconfining transition in $SU(3)$ gauge theory, traditionally interpreted through the Gross-Witten-Wadia (GWW) model as a sharp third-order phase transition in the large-$N_c$ limit, appears as a smooth crossover in lattice QCD. This…
We study a random matrix model which interpolates between the the singular values of the Gaussian unitary ensemble (GUE) and of the chiral Gaussian unitary ensemble (chGUE). This symmetry crossover is analogous to the one realized by the…
The Gardner transition is the transition that at mean-field level separates a stable glass phase from a marginally stable phase. This transition has similarities with the de Almeida-Thouless transition of spin glasses. We have studied a…
Spectral statistics of quantum chaotic systems are governed by random matrix universality. In many cases of interest, time-reversal symmetry selects the Gaussian Orthogonal Ensemble (GOE) as the relevant universality class. In holographic…
In this paper we study mesoscopic fluctuations for Dyson's Brownian motion with $\beta=2$. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from…
We study the McKean--Vlasov equation on the finite tori of length scale $L$ in $d$--dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in…
For the embedded Gaussian orthogonal ensemble (EGOE) of random matrices, the strength sums generated by a transition operator acting on an eigenstate vary with the excitation energy as the ratio of two Gaussians. This general result is…