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Register a new userUniversal Asymptotic Eigenvalue Distribution of Large $N$ Random Matrices --- A Direct Diagrammatic Proof to Marchenko-Pastur Law ---
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In random matrix theory, Marchenko-Pastur law states that random matrices with independent and identically distributed entries have a universal asymptotic eigenvalue distribution under large dimension limit, regardless of the choice of entry distribution. This law provides useful insight for physics research, because the large $N$ limit proved to be a very useful tool in various theoretical models. We present an alternative proof of Marchenko- Pastur law using Feynman diagrams, which is more familiar to the physics community. We also show that our direct diagrammatic approach can readily generalize to six types of restricted random matrices, which are not all covered by the original Marchenko-Pastur law.
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We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $Ntimes (N+ u)$ are correlated through a non-Hermiticity parameter $tauin[0,1]$. For general $ u=O(N)$ and $tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $mathcal{D}=begin{pmatrix} 0 & X_1 X_2^* & 0 end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostmans equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.
Motivated by supersymmetry breaking in matrix model formulations of superstrings, we present some concrete models, in which the supersymmetry is preserved for any finite $N$, but gets broken at infinite $N$, where $N$ is the rank of matrix variables. The models are defined as supersymmetric field theories coupled to some matrix models, and in the induced action obtained after integrating out the matrices, supersymmetry is spontaneously broken only when $N$ is infinity. In our models, the large value of $N$ gives a natural explanation for the origin of small parameters appearing in the field theories which trigger the supersymmetry breaking. In particular, in the case of the ORaifeartaigh model coupled to a certain supersymmetric matrix model, a nonsupersymmetric metastable vacuum appears near the origin of the field space, which is far from the position of the supersymmetric vacuum. We estimate its lifetime as a function of $N$.
A crucial result on the celebrated Sachdev-Ye-Kitaev model is that its large $N$ limit is dominated by melonic graphs. In this letter we offer a rigorous, diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs.
Using Majorana fermions and elementary mesons we find new massless quantum field theories with weakly interacting ultraviolet fixed points. We also find new classes of large N equivalences amongst SU, SO and Sp gauge theories with different types of matter fields and Yukawa interactions. Results include a triality of asymptotically safe theories and dualities between asymptotically free matter-gauge theories with identical fixed points, phase diagrams, and scaling exponents. Implications for conformal field theory and orbifold reductions are indicated.
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between $k$-tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with $k$-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.
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