Abstract: The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erdős-Renyi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present paper aims to extend that result to more-general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled, we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.

Abstract: We show that for each r ≥ 4 $$ r\ge 4 $$ , in a density range extending up to, and slightly beyond, the threshold for a K r $$ {K}_r $$ -factor, the copies of K r $$ {K}_r $$ in the random graph G ( n , p ) $$ G\left(n,p\right) $$ are randomly distributed, in the (one-sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus Jeff Kahn's recent asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem implies a corresponding bound for the threshold for G ( n , p ) $$ G\left(n,p\right) $$ to contain a K r $$ {K}_r $$ -factor. The case r = 3 $$ r=3 $$ is more difficult, and has been settled by Annika Heckel. We also prove a corresponding result for K r ( t ) $$ {K}_r^{(t)} $$ -factors in random t $$ t $$ -uniform hypergraphs, as well as (in some cases weaker) generalizations replacing K r $$ {K}_r $$ by certain other (hyper)graphs.

Abstract: Mossel and Ross raised the question of when a random colouring of a graph can be reconstructed from local information, namely the colourings (with multiplicity) of balls of given radius. In this paper, we are concerned with random $2$-colourings of the vertices of the $n$-dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter $\Omega(n)$ are required to reconstruct. However, the situation for random colourings is dramatically different: we show that almost every $2$-colouring can be reconstructed from the multiset of colourings of balls of radius $2$. Furthermore, we show that for $q \ge n^{2+\epsilon}$, almost every $q$-colouring can be reconstructed from the multiset of colourings of $1$-balls.

Abstract: Height functions of growing random surfaces are often conjectured to be superconcentrated, meaning that their variances grow sublinearly in time. This article introduces a new concept—called subroughness—meaning that there exist two distinct points such that the expected squared difference between the heights at these points grows sublinearly in time. The main result of the paper is that superconcentration is equivalent to subroughness in a class of growing random surfaces. The result is applied to establish superconcentration in a variant of the restricted solid-on-solid (RSOS) model and in a variant of the ballistic deposition model, and give new proofs of superconcentration in directed last-passage percolation and directed polymers.

Abstract: Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech-complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski–Weinberger and Bobrowski–Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: the corresponding common leading terms for the upper and lower thresholds differ being log ( n ) when M is closed and ( 2 − 2 / d ) log ( n ) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle occurring close to the boundary.

Abstract: Given a graphical degree sequence d = ( d 1 , … , d n ) $$ \mathbf{d}=\left({d}_1,\dots, {d}_n\right) $$ , let 𝒢 ( n , d ) denote a uniformly random graph on vertex set [ n ] $$ \left[n\right] $$ where vertex i $$ i $$ has degree d i $$ {d}_i $$ for every 1 ≤ i ≤ n $$ 1\le i\le n $$ . We give upper and lower bounds on the joint probability of an arbitrary set of edges in 𝒢 ( n , d ), and we link these probability estimates to the corresponding probabilities in the configuration model. Then we show that many existing results of 𝒢 ( n , d ) in the literature can be significantly improved with simpler proofs, by applying this new probabilistic tool. One example we give concerns the chromatic number of 𝒢 ( n , d ). In another application, we use these joint probabilities to study the connectivity of 𝒢 ( n , d ). Under some rather mild condition on d $$ \mathbf{d} $$ —in particular, if Δ 2 = o ( M ) $$ {\Delta}^2=o(M) $$ where Δ $$ \Delta $$ is the maximum component of d $$ \mathbf{d} $$ —we fully characterize the connectivity phase transition of 𝒢 ( n , d ). We also give sufficient conditions for 𝒢 ( n , d ) being connected when Δ $$ \Delta $$ is unrestricted.

Abstract: The Komlós conjecture suggests that for any vectors a 1 , … , a n ∈ B 2 m $$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_2^m $$ there exist x 1 , … , x n ∈ { − 1 , 1 } $$ {x}_1,\dots, {x}_n\in \left\{-1,1\right\} $$ so that ‖ ∑ i = 1 n x i a i ‖ ∞ ≤ O ( 1 ) $$ {\left\Vert {\sum}_{i=1}^n{x}_i{\boldsymbol{a}}_i\right\Vert}_{\infty}\le O(1) $$ . It is a natural extension to ask what ℓ q $$ {\ell}_q $$ -norm bound to expect for a 1 , … , a n ∈ B p m $$ {\boldsymbol{a}}_1,\dots, {\boldsymbol{a}}_n\in {B}_p^m $$ . We prove a tight partial coloring result for such vectors, implying a nearly tight full coloring bound. As a corollary, this implies a special case of Beck–Fiala's conjecture. We achieve this by showing that, for any δ > 0 $$ \delta >0 $$ , a symmetric convex body K ⊆ ℝ n $$ K\subseteq {\mathbb{R}}^n $$ with Gaussian measure at least e − δ n $$ {e}^{-\delta n} $$ admits a partial coloring. Previously this was known only for a small enough δ $$ \delta $$ . Additionally, we show that a hereditary volume bound suffices to provide such Gaussian measure lower bounds.

Abstract: Given a graph sequence $\{G_n\}_{n \geq 1}$ denote by $T_3(G_n)$ the number of monochromatic triangles in a uniformly random coloring of the vertices of $G_n$ with $c \geq 2$ colors. This arises as a generalization of the birthday paradox, where $G_n$ corresponds to a friendship network and $T_3(G_n)$ counts the number of triples of friends with matching birthdays. In this paper we prove a central limit theorem (CLT) for $T_3(G_n)$ with explicit error rates. The proof involves constructing a martingale difference sequence by carefully ordering the vertices of $G_n$, based on a certain combinatorial score function, and using a quantitive version of the martingale CLT. We then relate this error term to the well-known fourth moment phenomenon, which, interestingly, holds only when the number of colors $c \geq 5$. We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any $c \geq 2$, which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of $T_3(G_n)$, whenever $c \geq 5$. Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in Bhattacharya et al. (2017).

Abstract: We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson–Schensted–Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.

Abstract: We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of 𝔾 ( n , ω ( 1 ) / n ), using a palette of size n $$ n $$ , a.a.s. admits a rainbow copy of any given bounded-degree tree on at most ( 1 − ε ) n $$ \left(1-\varepsilon \right)n $$ vertices, where ε > 0 $$ \varepsilon >0 $$ is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded-degree almost-spanning prescribed trees in 𝔾 ( n , C / n ), where C > 0 $$ C>0 $$ is independent of n $$ n $$ . Given an n $$ n $$ -vertex graph G $$ G $$ with minimum degree at least δ n $$ \delta n $$ , where δ > 0 $$ \delta >0 $$ is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph G ∪ 𝔾 ( n , ω ( 1 ) / n ), using ( 1 + α ) n $$ \left(1+\alpha \right)n $$ colors, where α > 0 $$ \alpha >0 $$ is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that G ∪ 𝔾 ( n , C / n ), where C > 0 $$ C>0 $$ is independent of n $$ n $$ , a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with G $$ G $$ as above, we prove that a uniform coloring of G ∪ 𝔾 ( n , ω ( n − 2 ) ) using n − 1 $$ n-1 $$ colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

Abstract: We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic n $$ n $$ -vertex graph H $$ H $$ with δ ( H ) ≥ α n $$ \delta (H)\ge \alpha n $$ and a random d $$ d $$ -regular graph G $$ G $$ , for d ∈ { 1 , 2 } $$ d\in \left\{1,2\right\} $$ . When G $$ G $$ is a random 2-regular graph, we prove that a.a.s. H ∪ G $$ H\cup G $$ is pancyclic for all α ∈ ( 0 , 1 ] $$ \alpha \in \left(0,1\right] $$ , and also extend our result to a range of sublinear degrees. When G $$ G $$ is a random 1-regular graph, we prove that a.a.s. H ∪ G $$ H\cup G $$ is pancyclic for all α ∈ ( 2 − 1 , 1 ] $$ \alpha \in \left(\sqrt{2}-1,1\right] $$ , and this result is best possible. Furthermore, we show that this bound on δ ( H ) $$ \delta (H) $$ is only needed when H $$ H $$ is “far” from containing a perfect matching, as otherwise we can show results analogous to those for random 2-regular graphs. Our proofs provide polynomial-time algorithms to find cycles of any length.

Abstract: The Swendsen–Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to its global nature. We present optimal bounds on the convergence rate of the Swendsen–Wang algorithm for the complete d $$ d $$ -ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as variance mixing and entropy mixing imply Ω ( 1 ) $$ \Omega (1) $$ spectral gap and O ( log n ) $$ O\left(\log n\right) $$ mixing time, respectively, for the Swendsen–Wang dynamics on the d $$ d $$ -ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish Θ ( log n ) $$ \Theta \left(\log n\right) $$ mixing for the Swendsen–Wang dynamics for all boundary conditions throughout (and beyond) the tree uniqueness region. Our proofs feature a novel spectral view of the variance mixing condition and utilize recent work on block factorization of entropy.

Abstract: We show that for any d = d ( n ) with d 0 ( ϵ ) ≤ d = o ( n ) , with high probability, the size of a largest induced cycle in the random graph G ( n , d / n ) is ( 2 ± ϵ ) n d log d . This settles a long-standing open problem in random graph theory.

Abstract: Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok for designing deterministic approximation algorithms for various graph counting and related problems. An earlier method used for the same problem is one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply often coincide or nearly coincide. In this article we show that this is not a coincidence. We establish that if the interpolation method is valid for a family of graphs, then this family exhibits a form of the correlation decay property which is asymptotic strong spatial mixing at superlogarithmic distances. Our proof is based on a certain graph polynomial representation of the associated partition function. This representation is at the heart of the design of the polynomial time algorithms underlying the interpolation method itself. We conjecture that our result holds for all, and not just amenable graphs. Indeed this conjecture was recently confirmed by Regts. See the body of the article for details.

Abstract: A random graph model on a host graph H$$ H $$ is said to be 1‐independent if for every pair of vertex‐disjoint subsets A,B$$ A,B $$ of E(H)$$ E(H) $$ , the state of edges (absent or present) in A$$ A $$ is independent of the state of edges in B$$ B $$ . For an infinite connected graph H$$ H $$ , the 1‐independent critical percolation probability p1,c(H)$$ {p}_{1,c}(H) $$ is the infimum of the p∈[0,1]$$ p\in \left[0,1\right] $$ such that every 1‐independent random graph model on H$$ H $$ in which each edge is present with probability at least p$$ p $$ almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that p1,c(ℤd)$$ {p}_{1,c}\left({\mathbb{Z}}^d\right) $$ tends to a limit in [12,1]$$ \left[\frac{1}{2},1\right] $$ as d→∞$$ d\to \infty $$ , and they asked for the value of this limit. We make progress on a related problem by showing that limn→∞p1,c(ℤ2×Kn)=4−23=0.5358….$$ \underset{n\to \infty }{\lim }{p}_{1,c}\left({\mathbb{Z}}^2\times {K}_n\right)=4-2\sqrt{3}=0.5358\dots . $$In fact, we show that the equality above remains true if the sequence of complete graphs Kn$$ {K}_n $$ is replaced by a sequence of weakly pseudorandom graphs on n$$ n $$ vertices with average degree ω(logn)$$ \omega \left(\log n\right) $$ . We conjecture the answer to Balister and Bollobás's question is also 4−23$$ 4-2\sqrt{3} $$ .

Abstract: Let $G$ be an $n$-vertex graph, where $\delta(G) \geq \delta n$ for some $\delta := \delta(n)$. A result of Bohman, Frieze and Martin from 2003 asserts that if $\alpha(G) = O \left(\delta^2 n \right)$, then perturbing $G$ via the addition of $\omega \left(\frac{\log(1/\delta)}{\delta^3} \right)$ random edges, asymptotically almost surely (a.a.s. hereafter) results in a Hamiltonian graph. This bound on the size of the random perturbation is only tight when $\delta$ is independent of $n$ and deteriorates as to become uninformative when $\delta = \Omega \left(n^{-1/3} \right)$. We prove several improvements and extensions of the aforementioned result. First, keeping the bound on $\alpha(G)$ as above and allowing for $\delta = \Omega(n^{-1/3})$, we determine the correct order of magnitude of the number of random edges whose addition to $G$ a.a.s. results in a pancyclic graph. Our second result ventures into significantly sparser graphs $G$; it delivers an almost tight bound on the size of the random perturbation required to ensure pancyclicity a.a.s., assuming $\delta(G) = \Omega \left((\alpha(G) \log n)^2 \right)$ and $\alpha(G) \delta(G) = O(n)$. Assuming the correctness of Chv\'atal's toughness conjecture, allows for the mitigation of the condition $\alpha(G) = O \left(\delta^2 n \right)$ imposed above, by requiring $\alpha(G) = O(\delta(G))$ instead; our third result determines, for a wide range of values of $\delta(G)$, the correct order of magnitude of the size of the random perturbation required to ensure the a.a.s. pancyclicity of $G$. For the emergence of nearly spanning cycles, our fourth result determines, under milder conditions, the correct order of magnitude of the size of the random perturbation required to ensure that a.a.s. $G$ contains such a cycle.

Abstract: We establish the asymptotic normality of the dimension of large-size random Fishburn matrices by a complex-analytic approach. The corresponding dual problem of size distribution under large dimension is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to { 0 , 1 } $$ \left\{0,1\right\} $$ or nonnegative integers ℕ 0 $$ {\mathbb{N}}_0 $$ . The analytic saddle-point approach we apply, based on a powerful transformation for q $$ q $$ -series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.

Abstract: For graphs G , H $$ G,H $$ and a family of graphs ℱ $$ \mathcal{F} $$ , we write G → ( H , ℱ ) $$ G\to \left(H,\mathcal{F}\right) $$ to denote that every blue-red coloring of the edges of G $$ G $$ contains either a blue copy of H $$ H $$ , or a red copy of each F ∈ ℱ $$ F\in \mathcal{F} $$ . For integers n $$ n $$ and D $$ D $$ , let 𝒯 ( n , D ) denote the family of all trees with n $$ n $$ edges and maximum degree at most D $$ D $$ . We prove that for each r , D ⩾ 2 $$ r,D\geqslant 2 $$ , there exist constants C , C ′ > 0 $$ C,{C}^{\prime }>0 $$ such that if p ⩾ C n − 2 / ( r + 2 ) $$ p\geqslant C{n}^{-2/\left(r+2\right)} $$ and N ⩾ r n + C ′ / p $$ N\geqslant rn+{C}^{\prime }/p $$ , then G ( N , p ) → ( K r + 1 , 𝒯 ( n , D ) ) with high probability. This is a random version of a well-known result of Chvátal from 1977. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős–Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.

Abstract: For integers d ≥ 2 $$ d\ge 2 $$ and k ≥ d + 1 $$ k\ge d+1 $$ , a k $$ k $$ -hole in a set S $$ S $$ of points in general position in ℝ d $$ {\mathbb{R}}^d $$ is a k $$ k $$ -tuple of points from S $$ S $$ in convex position such that the interior of their convex hull does not contain any point from S $$ S $$ . For a convex body K ⊆ ℝ d $$ K\subseteq {\mathbb{R}}^d $$ of unit d $$ d $$ -dimensional volume, we study the expected number E H d , k K ( n ) $$ E{H}_{d,k}^K(n) $$ of k $$ k $$ -holes in a set of n $$ n $$ points drawn uniformly and independently at random from K $$ K $$ . We prove an asymptotically tight lower bound on E H d , k K ( n ) $$ E{H}_{d,k}^K(n) $$ by showing that, for all fixed integers d ≥ 2 $$ d\ge 2 $$ and k ≥ d + 1 $$ k\ge d+1 $$ , the number E H d , k K ( n ) $$ E{H}_{d,k}^K(n) $$ is at least Ω ( n d ) $$ \Omega \left({n}^d\right) $$ . For some small holes, we even determine the leading constant lim n → ∞ n − d E H d , k K ( n ) $$ {\lim}_{n\to \infty }{n}^{-d}E{H}_{d,k}^K(n) $$ exactly. We improve the currently best-known lower bound on lim n → ∞ n − d E H d , d + 1 K ( n ) $$ {\lim}_{n\to \infty }{n}^{-d}E{H}_{d,d+1}^K(n) $$ by Reitzner and Temesvari (2019). In the plane, we show that the constant lim n → ∞ n − 2 E H 2 , k K ( n ) $$ {\lim}_{n\to \infty }{n}^{-2}E{H}_{2,k}^K(n) $$ is independent of K $$ K $$ for every fixed k ≥ 3 $$ k\ge 3 $$ and we compute it exactly for k = 4 $$ k=4 $$ , improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche and by the authors.

Abstract: For G = G n , 1 / 2 $$ G={G}_{n,1/2} $$ , the Erdős–Renyi random graph, let X n $$ {X}_n $$ be the random variable representing the number of distinct partitions of V ( G ) $$ V(G) $$ into sets A 1 , … , A q $$ {A}_1,\dots, {A}_q $$ so that the degree of each vertex in G [ A i ] $$ G\left[{A}_i\right] $$ is divisible by q $$ q $$ for all i ∈ [ q ] $$ i\in \left[q\right] $$ . We prove that if q ≥ 3 $$ q\ge 3 $$ is odd then X n → d Po ( 1 / q ! ) $$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left(1/q!\right) $$ , and if q ≥ 4 $$ q\ge 4 $$ is even then X n → d Po ( 2 q / q ! ) $$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left({2}^q/q!\right) $$ . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G [ A i ] $$ G\left[{A}_i\right] $$ to be congruent to x i $$ {x}_i $$ modulo q $$ q $$ for each i ∈ [ q ] $$ i\in \left[q\right] $$ , where the residues x i $$ {x}_i $$ may be chosen freely. For q = 2 $$ q=2 $$ , the distribution is not asymptotically Poisson, but it can be determined explicitly.

Abstract: Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, \[ 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} \] for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum $N$ such that any $2$-edge-coloring of the Cartesian product $K_N \square K_N$ contains either a red rectangle or a blue $K_n$?

Abstract: We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.

Abstract: We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of"unusual"topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

Abstract: How can we approximate sparse graphs and sequences of sparse graphs (with unbounded average degree)? We consider convergence in the first k moments of the graph spectrum (equivalent to the numbers of closed k -walks) appropriately normalized. We introduce a simple random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The random overlapping communities (ROC) model is specified by a distribution on pairs ( s , q ) , s ∈ ℤ + , q ∈ ( 0 , 1 ] . A graph on n vertices with average degree d is generated by repeatedly picking pairs ( s , q ) from the distribution, adding an Erdős–Rényi random graph of edge density q on a subset of vertices chosen by including each vertex with probability s / n , and repeating this process so that the expected degree is d . We also show that ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real-world networks.

Abstract: We study cycle counts in permutations of $1,\dots,n$ drawn at random according to the Mallows distribution. Under this distribution, each permutation $\pi \in S_n$ is selected with probability proportional to $q^{\text{inv}(\pi)}$, where $q>0$ is a parameter and $\text{inv}(\pi)$ denotes the number of inversions of $\pi$. For $\ell$ fixed, we study the vector $(C_1(\Pi_n),\dots,C_\ell(\Pi_n))$ where $C_i(\pi)$ denotes the number of cycles of length $i$ in $\pi$ and $\Pi_n$ is sampled according to the Mallows distribution. Here we show that if $01$ there is striking difference between the behaviour of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behaviour depends on the parity of $n$ when $q>1$. Both $(C_1(\Pi_{2n}),C_3(\Pi_{2n}),\dots)$ and $(C_1(\Pi_{2n+1}),C_3(\Pi_{2n+1}),\dots)$ have discrete limiting distributions -- they do not need to be renormalized -- but the two limiting distributions are distinct for all $q>1$. We describe these limiting distributions in terms of Gnedin and Olshanski's bi-infinite extension of the Mallows model. We also investigate these limiting distributions, and study the behaviour of the constants involved in the Gaussian limit laws. We for example show that as $q\downarrow 1$ the expected number of 1-cycles tends to $1/2$ -- which, curiously, differs from the value corresponding to $q=1$. In addition we exhibit an interesting"oscillating"behaviour in the limiting probability measures for $q>1$ and $n$ odd versus $n$ even.

Abstract: We determine the asymptotic normalized rank of a random matrix A $$ \boldsymbol{A} $$ over an arbitrary field with prescribed numbers of nonzero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.

Abstract: The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of n$$ n $$ points in the unit square [0,1]2$$ {\left[0,1\right]}^2 $$ that maximizes the smallest area of a triangle formed by three of those points. This problem has natural generalizations. For an integer k≥3$$ k\ge 3 $$ and a set 𝒫 of n$$ n $$ points in [0,1]2$$ {\left[0,1\right]}^2 $$ , let Ak(𝒫) be the minimum area of the convex hull of k$$ k $$ points in 𝒫 . Here, instead of considering the supremum of Ak(𝒫) over all such choices of 𝒫 , we consider its average value, Δ˜k(n)$$ {\tilde{\Delta}}_k(n) $$ , when the n$$ n $$ points in 𝒫 are chosen independently and uniformly at random in [0,1]2$$ {\left[0,1\right]}^2 $$ . We prove that Δ˜k(n)=Θn−kk−2$$ {\tilde{\Delta}}_k(n)=\Theta \left({n}^{\frac{-k}{k-2}}\right) $$ , for every fixed k≥3$$ k\ge 3 $$ .

Abstract: We study the number c n ( N ) $$ {c}_n^{(N)} $$ of n $$ n $$ -step self-avoiding walks on the N $$ N $$ -dimensional hypercube, and identify an N $$ N $$ -dependent connective constant μ N $$ {\mu}_N $$ and amplitude A N $$ {A}_N $$ such that c n ( N ) $$ {c}_n^{(N)} $$ is O ( μ N n ) $$ O\left({\mu}_N^n\right) $$ for all n $$ n $$ and N $$ N $$ , and is asymptotically A N μ N n $$ {A}_N{\mu}_N^n $$ as long as n ≤ 2 p N $$ n\le {2}^{pN} $$ for any fixed p < 1 2 $$ p<\frac{1}{2} $$ . We refer to the regime n ≪ 2 N / 2 $$ n\ll {2}^{N/2} $$ as the dilute phase. We discuss conjectures concerning different behaviors of c n ( N ) $$ {c}_n^{(N)} $$ when n $$ n $$ reaches and exceeds 2 N / 2 $$ {2}^{N/2} $$ , corresponding to a critical window and a dense phase. In addition, we prove that the connective constant has an asymptotic expansion to all orders in N − 1 $$ {N}^{-1} $$ , with integer coefficients, and we compute the first five coefficients μ N = N − 1 − N − 1 − 4 N − 2 − 26 N − 3 + O ( N − 4 ) $$ {\mu}_N=N-1-{N}^{-1}-4{N}^{-2}-26{N}^{-3}+O\left({N}^{-4}\right) $$ . The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.