We consider the random‐cluster model (RCM) on with parameters p ∈(0,1) and q  ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p ) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of

Two simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile networks. We consider two independent walkers moving with constant speed on the discrete or continuous circle, and changing directions at independent geometric (respectively, exponential) times. One of the walkers carries a message that wishes to

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function f :N→[0,1] that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ . For this particular class, we prove

Consider the Aldous Markov chain on the space of rooted binary trees with n labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix 1 ≤ k

A 1‐factorization of a graph G is a collection of edge‐disjoint perfect matchings whose union is E (G ). In this paper, we prove that for any ϵ >0, an (n ,d ,λ )‐graph G admits a 1‐factorization provided that n is even, C 0 ≤ d  ≤ n −1 (where C 0=C 0(ϵ ) is a constant depending only on ϵ ), and λ  ≤ d 1−ϵ . In particular, since (as is well known) a typical random d ‐regular graph G n ,d is such a graph

Koiran's real τ‐conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial lower bounds for the arithmetic circuit size of the permanent. We confirm the conjecture

We study site percolation models on planar lattices including the [m,4,n,4] lattice and the square tilings on the Euclidean plane () or the hyperbolic plane (), satisfying certain local constraints on degree‐4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of or . In particular, we obtain a description of the numbers

We sharpen run‐time analysis for algorithms under the partial rejection sampling framework. Our method yields improved bounds for: the cluster‐popping algorithm for approximating all‐terminal network reliability; the cycle‐popping algorithm for sampling rooted spanning trees; and the sink‐popping algorithm for sampling sink‐free orientations. In all three applications, our bounds are not only tight

Let A be an n×n random matrix with independent rows R1(A),…,Rn(A), and assume that for any i ≤ n and any three‐dimensional linear subspace the orthogonal projection of Ri(A) onto F has distribution density satisfying (x∈F) for some constant C1>0. We show that for any fixed n×n real matrix M we have (1) where C′>0 is a universal constant. In particular, the above result holds if the rows of A are independent

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on Rd with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψτ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψτ is hyperuniform and number rigid, quite

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ϵ>0, we show that the expected absorption time on an n‐vertex graph is o(n3+ϵ). Specifically

We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the

We study a family of directed random graphs whose arcs are sampled independently of each other, and are present in the graph with a probability that depends on the attributes of the vertices involved. In particular, this family of models includes as special cases the directed versions of the Erdős‐Rényi model, graphs with given expected degrees, the generalized random graph, and the Poissonian random

We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting r vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant α. A new vertex

Let be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children u and v such that the subtrees rooted at u and v belong to it. Let p be the probability that a Galton‐Watson tree falls in . The metaproperty makes p satisfy a fixed‐point equation, which can have multiple solutions

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere Sd in , and a pair of vertices is connected if the Euclidean distance between them is at least 2−ϵ. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ϵ>0 is small and fixed, then the chromatic number is d+2 with high probability. We show

In this paper, we introduce a model of depth‐weighted random recursive trees, created by recursively joining a new leaf to an existing vertex . In this model, the probability of choosing depends on its depth in the tree. In particular, we assume that there is a function such that if has depth then its probability of being chosen is proportional to . We consider the expected value of the diameter of

We study the asymptotic behavior of the maximal multiplicity Mn = Mn(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with WeW = n. We show that, over subsequences {nk}k ≥ 1 of the sequence of the natural

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact

Let Xv for v∈V be a family of n iid uniform points in the square . Suppose first that we are given the random geometric graph , where vertices u and v are adjacent when the Euclidean distance dE(Xu,Xv) is at most r. Let n3/14≪r≪n1/2. Given G (without geometric information), in polynomial time we can with high probability approximately reconstruct the hidden embedding, in the sense that “up to symmetries

Consider an eigenvector of the adjacency matrix of a G(n,p) graph. A nodal domain is a connected component of the set of vertices where this eigenvector has a constant sign. It is known that with high probability, there are exactly two nodal domains for each eigenvector corresponding to a nonleading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately

A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element U. The value of the root is the key quantity of interest in general. In this study, all node values and function values are in a finite set S. In this note

Motivated by the Komlós conjecture in combinatorial discrepancy, we study the discrepancy of random matrices with m rows and n independent columns drawn from a bounded lattice random variable. We prove that for n at least polynomial in m, with high probability the ℓ∞‐discrepancy is at most twice the ℓ∞‐covering radius of the integer span of the support of the random variable. Applying this result to

For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in G. The modularity q∗(G) of the graph G is defined to be the maximum over all vertex partitions of the modularity score, and satisfies 0 ≤ q∗(G)<1. Modularity is at the heart of the most popular algorithms for community detection. We investigate

We present an algorithm CRE, which either finds a Hamilton cycle in a graph G or determines that there is no such cycle in the graph. The algorithm's expected running time over input distribution G∼G(n,p) is (1+o(1))n/p, the optimal possible expected time, for . This improves upon previous results on this problem due to Gurevich and Shelah, and to Thomason.

Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent‐entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any subset of its coordinates carries an appropriate proportion of its mass. Our results hold for random matrices with genuinely complex as well as real entries. As an application

The iterative absorption method has recently led to major progress in the area of (hyper‐)graph decompositions. Among other results, a new proof of the existence conjecture for combinatorial designs, and some generalizations, was obtained. Here, we illustrate the method by investigating triangle decompositions: We give a simple proof that a triangle‐divisible graph of large minimum degree has a triangle

A recent result of Condon, Kim, Kühn, and Osthus implies that for any , an n‐vertex almost r‐regular graph G has an approximate decomposition into any collections of n‐vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost αn‐regular graph G with any α>0 and a collection of bounded degree trees on at most (1−o(1))n vertices if G does not contain large bipartite

The Lovász local lemma (LLL) is a probabilistic tool to generate combinatorial structures with good “local” properties. The “LLL‐distribution” further shows that these structures have good global properties in expectation. The seminal algorithm of Moser and Tardos turned the simplest, variable‐based form of the LLL into an efficient algorithm; this has since been extended to other probability spaces

For positive integers r>ℓ, an r‐uniform hypergraph is called an ℓ‐cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of r consecutive vertices, and such that every pair of consecutive edges (in the natural ordering of the edges) intersect in precisely ℓ vertices; such cycles are said to be linear when ℓ=1, and nonlinear when ℓ>1. We determine the sharp threshold

Given a graph Γn=(V,E) on n vertices and m edges, we define the Erdős‐Rényi graph process with host Γn as follows. A permutation e1,…,em of E is chosen uniformly at random, and for t ≤ m we let Γn,t=(V,{e1,…,et}). Suppose the minimum degree of Γn is δ(Γn) ≥ (1/2+ε)n for some constant ε>0. Then with high probability (An event holds with high probability (whp) if as n→∞.), Γn,t becomes Hamiltonian at

Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that, assuming essentially a bounded second moment of the degree distribution, this construction with the simplest types of switchings yields a simple random graph with

We prove a rainbow version of the blow‐up lemma of Komlós, Sárközy, and Szemerédi for μn‐bounded edge colorings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow‐up lemma can be used to transfer the bandwidth theorem of Böttcher, Schacht, and Taraz to the rainbow setting. It can also be employed as a tool beyond the

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1,2,…,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each γ>0 and for all n>n0(γ). Suppose that (i) the maximum degree of T is bounded by ), and (ii) the

Let Ωq=Ωq(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γq be the graph with vertex set Ωq where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=Hn,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γq is connected if d is sufficiently large and . This is optimal up to the first

Bollobás, Reed, and Thomason proved every 3‐uniform hypergraph ℋ with m edges has a vertex‐partition V()=V1⊔V2⊔V3 such that each part meets at least edges, later improved to 0.6m by Halsegrave and improved asymptotically to 0.65m+o(m) by Ma and Yu. We improve this asymptotic bound to , which is best possible up to the error term, resolving a special case of a conjecture of Bollobás and Scott.

We consider the shotgun assembly problem for a random jigsaw puzzle, introduced by Mossel and Ross (2015). Their model consists of a puzzle—an n×n grid, where each vertex is viewed as a center of a piece. Each of the four edges adjacent to a vertex is assigned one of q colors (corresponding to “jigs,” or cut shapes) uniformly at random. Unique assembly refers to there being only one puzzle (the original

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly

We study approximate decompositions of edge‐colored quasirandom graphs into rainbow spanning structures: an edge‐coloring of a graph is locally ‐bounded if every vertex is incident to at most edges of each color, and is (globally) ‐bounded if every color appears at most times. Our results imply the existence of: (1) approximate decompositions of properly edge‐colored into rainbow almost‐spanning cycles;

Let be the orientable surface of genus and denote by the class of all graphs on vertex set with edges embeddable on . We prove that the component structure of a graph chosen uniformly at random from features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős‐Rényi random graph chosen uniformly at random from all graphs with vertex set and edges. It

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability

We consider supercritical bond percolation on a family of high‐girth ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any . It was further shown in

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ

In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k-dimensional Laplacian for 1 ≤ k ≤ d. We study an example of random walks on simplicial complexes in the context of a semi-supervised learning problem. Specifically, we consider

A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the k-SAT problem or the Ising model. Such models naturally induce a Gibbs

For any set Ω of non-negative integers such that { 0 , 1 } ⊊ Ω , we consider a random Ω-k-tree G n,k that is uniformly selected from all connected k-trees of (n + k) vertices such that the number of (k + 1)-cliques that contain any fixed k-clique belongs to Ω. We prove that Gn,k, scaled by ( k H k σ Ω ) / ( 2 n ) where H k is the kth harmonic number and σ Ω > 0, converges to the continuum random tree