Related papers: Upper Tails of Subgraph Counts in Sparse Regular G…

Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+\delta$. Going beyond the Erd\H{o}s-R\'enyi model, we establish…

The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erd\H{o}s--R\'enyi random graph exceeds its expectation by some constant factor. There has been much exciting…

The upper tail problem in a sparse Erd\H{o}s-R\'enyi graph asks for the probability that the number of copies of some fixed subgraph exceeds its expected value by a constant factor. We study the analogous problem for oriented subgraphs in…

We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erd\H{o}s-R\'enyi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/\Delta}…

The upper tail problem in the Erd\H{o}s--R\'enyi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$. Chatterjee and Dembo…

We study the typical structure of a sparse Erd\H{o}s--R\'enyi random graph conditioned on the lower tail subgraph count event. We show that in certain regimes, a typical graph sampled from the conditional distribution resembles the entropy…

For any fixed simple graph $H=(V,E)$ and any fixed $u>0$, we establish the leading order of the exponential rate function for the probability that the number of copies of $H$ in the Erd\H{o}s--R\'enyi graph $G(n,p)$ exceeds its expectation…

In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erd\H{o}s-R\'enyi random graph $\mathcal{G}_{n,p}$, in the regime of $p$ where the edge of the spectrum is no…

We study the lower tail large deviation problem for subgraph counts in a random graph. Let $X_H$ denote the number of copies of $H$ in an Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail…

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to…

The "infamous upper tail problem" for $r$-uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph $H$ in a large binomial $r$-uniform hypergraph $\boldsymbol{G}$ exceeds its expectation by a…

For a $\Delta$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$, an Erd\H{o}s-R\'{e}nyi graph on $n$ vertices with edge probability $p$,…

The upper tail problem for the largest eigenvalue of the Erd\H{o}s--R\'enyi random graph $\mathcal{G}_{n,p}$ is to estimate the probability that the largest eigenvalue of the adjacency matrix of $\mathcal{G}_{n,p}$ exceeds its typical value…

Let $X$ count the number of $r$-stars in the random binomial graph $\mathbb{G}(n,p)$. We determine, for fixed $r$ and $\varepsilon > 0$, the asymptotics of $\log \mathbb{P}(X \ge (1 + \varepsilon)\mathbb{E} X)$ assuming only $\mathbb{E} X…

Let $N$ be the number of triangles in an Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that $N$ weakly converges to the Poisson distribution with mean…

With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}s-R\'enyi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$, we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k}…

For any $S\subset [n]$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is a given graph $H$ on the vertex set $S$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{{\bf…

We consider an Erd\H{o}s-R\'{e}nyi graph $\mathbb{G}(n,p)$ on $n$ vertices with edge probability $p$ such that \[ \sqrt{\frac{\log n}{\log \log n}} \ll np \le n^{1/2-o(1)}, \label{eq:abs} \tag{$\dagger$} \] and derive the upper tail large…

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and…

Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4-cycle in the…