In the binomial random graph @math , when @math changes from @math (subcritical case) to @math and then to @math (supercritical case) for @math , with high probability the order of the largest component increases smoothly from @math to @math and then to @math . As a natural generalisation of random graphs and connectedness, we consider the binomial random @math -uniform hypergraph @math (where each @math -tuple of vertices is present as a hyperedge with probability @math independently) and the following notion of high-order connectedness. Given an integer @math , two sets of @math vertices are called if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least @math vertices. A @math -connected component is a maximal collection of pairwise @math -connected @math -tuples of vertices. Recently, the threshold for the appearance of the giant @math -connected component in @math and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure, order, and size of the largest @math -connected components, with the help of a certain class of hypertrees' and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant @math -connected component.
The case @math of @math -tuple-connectedness corresponds to vertex-connectedness of @math and is the most studied among the higher-dimensional analogues of the phase transition in @math . Enumeration results for the asymptotic number of @math -connected @math -uniform hypergraphs were obtained by Karo 'nski and uczak @cite_11 , later improved by Andriamampianina and Ravelomanana @cite_1 via enumerative techniques. The threshold for the emergence of the giant @math -component, @math , was first determined by Schmidt-Pruzan and Shamir @cite_25 . Subsequently, Karo 'nski and uczak @cite_3 studied the distribution of the order of the largest component in the early supercritical regime. The studied range was extended by Ravelomanana and Rijamamy @cite_15 , although they only computed the expected order of the largest component and not its distribution. Behrisch, Coja-Oghlan, and Kang @cite_10 @cite_5 provided central and local limit theorems for @math , with @math arbitrarily small but fixed, while more recently Bollob 'as and Riordan @cite_2 showed that the distribution of the order of the largest component tends to a normal distribution for every @math .
@cite_1 @cite_3 @cite_2 @cite_5 @cite_15 @cite_10 @cite_25 @cite_11
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The theme of this work is an "inside-out" approach to the enumeration of graphs. It is based on a well-known decomposition of a graph into its 2-core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number of 2-cores with a given number of vertices and edges, we solve the corresponding enumeration problem for the connected 2-cores. For a subrange of the parameters, we also enumerate those 2-cores by using a deeper inside-out notion of a kernel of a connected 2-core.Using this enumeration result in combination with Caley's formula for forests, we obtain an alternative and simpler proof of the asymptotic formula of Bender, Canfield and McKay for the number of connected graphs with n vertices and m edges, with improved error estimate for a range of m values.As another application, we study the limit joint distribution of three parameters of the giant component of a random graph with n vertices in the supercritical phase, when the difference between average vertex degree and 1 far exceeds n-1 3. The three parameters are defined in terms of the 2-core of the giant component, i.e. its largest subgraph of minimum degree 2 or more. They are the number of vertices in the 2-core, the excess (#edges - #vertices) of the 2-core, and the number of vertices not in the 2-core. We show that the limit distribution is jointly Gaussian throughout the whole supercritical phase. In particular, for the first time, the 2-core size is shown to be asymptotically normal, in the widest possible range of the average vertex degree. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. Consider the random graph on n vertices 1,...,n. Each vertex i is assigned a type x(i) with x(1),...,x(n) being independent identically distributed as a nonnegative random variable X. We assume that EX3 < infinity. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability min 1, x(i)x(j) n (1 + a n(1 3)) . We study the critical phase, which is known to take place when EX2 = 1. We prove that normalized by n(-2 3) the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient root EXEX3 and drift a - EX3 EX s. In particular, we conclude that the size of the largest connected component is of order n(2 3). (c) 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486-539, 2013 (Less)