The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes "without replacement" [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact "tree-like" random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $(\tau-2)/(\tau-3)$ a.s., in stark contrast to the Erd\H{o}s-R\'{e}nyi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$.Comment: 71 pages, 5 figures, To appear in Probability Theory and Related Field

Universality for critical heavy-tailed network models: Metric structure of maximal components

We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen [Probab. Theory Relat. Fields 2018]. We develop general principles under which the identical scaling limits as the rank-one case can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.Comment: Final published version. 47 pages, 6 figure

Euler Number and Percolation Threshold on a Square Lattice with Diagonal Connection Probability and Revisiting the Island-Mainland Transition

We report some novel properties of a square lattice filled with white sites, randomly occupied by black sites (with probability $p$). We consider connections up to second nearest neighbours, according to the following rule. Edge-sharing sites, i.e. nearest neighbours of similar type are always considered to belong to the same cluster. A pair of black corner-sharing sites, i.e. second nearest neighbours may form a 'cross-connection' with a pair of white corner-sharing sites. In this case assigning connected status to both pairs simultaneously, makes the system quasi-three dimensional, with intertwined black and white clusters. The two-dimensional character of the system is preserved by considering the black diagonal pair to be connected with a probability $q$, in which case the crossing white pair of sites are deemed disjoint. If the black pair is disjoint, the white pair is considered connected. In this scenario we investigate (i) the variation of the Euler number $\chi(p) \ [=N_B(p)-N_W(p)]$ versus $p$ graph for varying $q$, (ii) variation of the site percolation threshold with $q$ and (iii) size distribution of the black clusters for varying $p$, when $q=0.5$. Here $N_B$ is the number of black clusters and $N_W$ is the number of white clusters, at a certain probability $p$. We also discuss the earlier proposed 'Island-Mainland' transition (Khatun, T., Dutta, T. & Tarafdar, S. Eur. Phys. J. B (2017) 90: 213) and show mathematically that the proposed transition is not, in fact, a critical phase transition and does not survive finite size scaling. It is also explained mathematically why clusters of size 1 are always the most numerous

Limiting spectral distribution of sample autocovariance matrices

We show that the empirical spectral distribution (ESD) of the sample autocovariance matrix (ACVM) converges as the dimension increases, when the time series is a linear process with reasonable restriction on the coefficients. The limit does not depend on the distribution of the underlying driving i.i.d. sequence and its support is unbounded. This limit does not coincide with the spectral distribution of the theoretical ACVM. However, it does so if we consider a suitably tapered version of the sample ACVM. For banded sample ACVM the limit has unbounded support as long as the number of non-zero diagonals in proportion to the dimension of the matrix is bounded away from zero. If this ratio tends to zero, then the limit exists and again coincides with the spectral distribution of the theoretical ACVM. Finally, we also study the LSD of a naturally modified version of the ACVM which is not non-negative definite.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ520 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Scaling limits and universality: Critical percolation on weighted graphs converging to an L3L^3 graphon

We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdos-Renyi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$ graphon. In particular, we define a notion of the critical window in this setting. The $L^3$ assumption ensures that the model is in the Erdos-Renyi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel's RGIV model [9] inside the critical window. Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs [52]. In [10], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Rucinski and Wormald's random graph process with degree restrictions [56] provided an additional technical condition about the barely subcritical behavior of this model can be proved.Comment: 65 pages, 1 figure, the universality principle (Theorem 3.4) from arXiv:1411.3417 has now been included in this paper. v2: minor change

Global lower mass-bound for critical configuration models in the heavy-tailed regime

We establish the global lower mass-bound property for the largest connected components in the critical window for the configuration model when the degree distribution has an infinite third moment. The scaling limit of the critical percolation clusters, viewed as measured metric spaces, was established in [7] with respect to the Gromov-weak topology. Our result extends those scaling limit results to the stronger Gromov-Hausdorff-Prokhorov topology under slightly stronger assumptions on the degree distribution. This implies the distributional convergence of global functionals such as the diameter of the largest critical components. Further, our result gives a sufficient condition for compactness of the random metric spaces that arise as scaling limits of critical clusters in the heavy-tailed regime.Comment: 25 page