Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex degree and form very large numbers of simple polygons (triangles or squares). The models avoid the collapse phenomena that plague naive graph Hamiltonians based on triangle or square counts. More than that, statistically significant numbers of other geometric primitives (small pieces of regular lattices, cubes) emerge in our ensemble, even though they are not in any way explicitly pre-programmed into the formulation of the graph Hamiltonian, which only depends on properties of paths of length 2. While much of our motivation comes from hopes to construct a graph-based theory of random geometry (Euclidean quantum gravity), our presentation is completely self-contained within the context of exponential random graph theory, and the range of potential applications is considerably more broad.

We consider an Erdős–Rényi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N → ∞ one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N → ∞ in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c = 4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation 2 / c 3 . At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N → ∞ distribution suggested by the numerics.

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve p_zrs(λ) that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetry-based techniques.&#xD;&#xD;We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with N vertices of mean degree c. In the regime 1<<c<<N , the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of p_zrs (λ), centered at c with width ∼sqrt(c). At smaller c, this curve receives corrections in powers of 1/sqrt(c) accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large c limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.&#xD;

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