Abstract. We consider the Potts model with q colors on a sequence of weighted graphs with adjacency matrices An, allowing for both positive and negative weights. Under a mild regularity condition on An we show that the mean-field prediction for the log partition function is asymptotically correct, whenever tr(A 2 n ) = o(n). In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to +∞. Using this, we establish the universality of the limiting log partition function of the ferromagnetic Potts model for a sequence of asymptotically regular graphs, and that of the Ising model for bi-regular bipartite graphs in both ferromagnetic and anti-ferromagnetic domain. We also derive a large deviation principle for the empirical measure of the colors for the Potts model on asymptotically regular graphs.

Abstract. We consider a class of sparse random matrices of the form An = (ξi,jδi,j ) n i,j=1 , where {ξi,j } are i.i.d. centered random variables, and {δi,j } are i.i.d. Bernoulli random variables taking value 1 with probability pn, and prove a quantitative estimate on the smallest singular value for pn = Ω( log n n ), under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For pn = Ω(n −α ) with some α ∈ (0, 1), we deduce that the condition number of An is of order n with probability tending to one under the optimal moment assumption on {ξi,j}. This in particular, extends a conjecture of von Neumann about the condition number to sparse random matrices with heavy-tailed entries. In the case that the random variables {ξi,j } are i.i.d. sub-Gaussian, we further show that a sparse random matrix is singular with probability at most exp(−cnpn) whenever pn is above the critical threshold pn = Ω( log n n ). The results also extend to the case when {ξi,j } have a non-zero mean. We further find quantitative estimates on the smallest singular value of the adjacency matrix of a directed Erdős-Réyni graph whenever its edge connectivity probability is above the critical threshold Ω( log n n ).

Let P 1 n , . . . , P d n be n × n permutation matrices drawn independently and uniformly at random, and set S d n := d ℓ=1 P ℓ n . We show that if log 12 n/(log log n) 4 ≤ d = O(n), then the empirical spectral distribution of S d n / √ d converges weakly to the circular law in probability as n → ∞. 1 1.1. Background: esd's for non-normal matrices. The study of the esd for random Hermitian matrices can be traced back to Wigner [42,43] who showed that the esd's of n × n Hermitian matrices with i.i.d. centered entries of variance 1/n (upper diagonal) satisfying appropriate moment bounds (e.g., Gaussian) converge to the semicircle distribution. The conditions on finiteness of moments were removed in subsequent work, see e.g. [5,34] and the references therein. We refer to the texts [30,21,39,3,5] for further background and a historical perspective.Wigner's proof employed the method of moments: one notes that the moments of the semicircle law determine it, and then one computes by combinatorial means the expectation (and variance) of the trace of powers of the matrix. This method (as well as related methods based on evaluating the Stieltjes transform of the esd) fails for non-normal matrices since moments do not determine the esd.

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let M N be a deterministic N × N matrix, and let G N be a complex Ginibre matrix. We consider the matrix M N = M N + N −γ G N , where γ > 1/2. With L N the empirical measure of eigenvalues of M N , we provide a general deterministic equivalence theorem that ties L N to the singular values of z − M N , with z ∈ C. We then compute the limit of L N when M N is an upper triangular Toeplitz matrix of finite symbol: if M N = d i=0 a i J i where d is fixed, a i ∈ C are deterministic scalars and J is the nilpotent matrix J(i, j) = 1 j=i+1 , then L N converges, as N → ∞, to the law of d i=0 a i U i where U is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when d = 1, also of i.i.d. entries on the diagonals in M N .