How does the core sit inside the mantle?

The“giant component” has remained a guiding theme in the theory of random graphs ever since the seminal paper of Erdős and Rényi [Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960) 17–61]. Because for any k ≥ 3 the k-core, defined as the (unique) maximal subgraph of minimum degree k, is identical to the largest kconnected subgraph of the random graph w.h.p., the k-core is perhaps the most natural generalisation of the “giant component”. Pittel, Wormald and Spencer were the first to determine the precise threshold dk beyond which the k-core Ck (G) of G =G(n,d/n) with d > 0 fixed is non-empty w.h.p. [Journal of Combinatorial Theory, Series B 67 (1996) 111–151]. Specifically, for any k ≥ 3 there is a function ψk : (0,∞) → [0,1] such that for any d ∈ (0,∞) \ {dk } the sequence (n−1|Ck (G)|)n converges to ψk (d) in probability. The aim of the present paper is to enhance the branching process perspective of the k-core problem pointed out in their paper. More specifically, we are concerned with the following question. Fix k ≥ 3, d > dk and let s > 0 be an integer. Generate a random graph G and mark each vertex according to σk,G : V (G) → {0,1} , v 7→ 1 {v ∈Ck (G)} . For a vertex v let G v denote its component. Now, pick a vertex v uniformly at random and let ∂s [G v , v ,σk,G v ] denote the isomorphism class of the finite rooted {0,1}-marked graph obtained by deleting all vertices at distance greater than s from v from G v . Our aim is to determine the distribution of ∂s [G v , v ,σk,G v ]. To accomodate the non-trivial correlations between the k-core and the “mantle” (i.e., the vertices outside the core) we introduce a Galton-Watson process T̂ (d ,k, p) that posseses five vertex types, denoted by 000, 001, 010, 110, 111. Setting q = q(d ,k, p) = P[Po(d p) = k −1|Po(d p) ≥ k −1] , we let p000 = 1−p, p010 = pq , p110 = p(1− q). The process starts with a single vertex, whose type is chosen from {000,010,111} according to the distribution (p000, p010, p111). Subsequently, each vertex of type z1z2z3 spawns a random number of vertices of each type. The offspring distributions are defined by the generating functions gz1z2z2 (x) detailed in Figure 1, where x = (x000, x001, x010, x110, x111) and q̄ = q̄(d ,k, p) = P [ Po(d p) = k −2|Po(d p) ≤ k −2] . Let T (d ,k, p) signify the random rooted {0,1}-marked tree obtained by giving mark 0 to all vertices of type 000, 001 or 010, and mark 1 to all others. g000(x) = exp(d(1−p)x000) ∑k−2 h=0(d p) h (qx010 + (1−q)x110) /h! ∑k−2 h=0(d p) h /h! , g001(x) = q̄ ( exp(d(1−p)x001) ( qx010 + (1−q)x110 )k−2) + (1− q̄) ( exp(d(1−p)x000) ∑k−3 h=0(d p) h (qx010 + (1−q)x110) /h! ∑k−3 h=0(d p) h /h! ) , g010(x) = exp(d(1−p)x001) ( qx010 + (1−q)x110 )k−1 , g110(x) = exp(d(1−p)x001) ∑ h≥k (d px111) /h! ∑ h≥k (d p)h /h! , g111(x) = exp(d(1−p)x001) ∑ h≥k−1(d px111) /h! ∑ h≥k−1(d p)h /h! . FIGURE 1. The generating functions gz1z2z3 (x). Theorem. Assume that k ≥ 3 and d > dk . Let s ≥ 0 be an integer and let τ be a rooted {0,1}-marked tree. Moreover, let p∗ be the largest fixed point of φd ,k : [0,1] → [0,1], p 7→P [ Po(d p) ≥ k −1] . Then 1 n ∑ v∈V (G) 1 { ∂ [G , v,σk,G v ] = ∂ [τ] } converges to P [ ∂ [T (d ,k, p∗)] = ∂ [τ]] in probability. This is joint work with Amin Coja-Oghlan, Oliver Cooley and Mihyun Kang.