arXiv subjects
Publications and source records attributed to Yufei Zhao.
We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, Rödl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.
We give a short elementary proof of the main theorem in the paper "Differential calculus on graphon space" by Diao et al. (JCTA 2015), which says that any graphon parameters whose $(N+1)$-th derivatives all vanish must be a linear combination of homomorphism densities $t(H, -)$ over graphs $H$ on at most $N$ edges.
The following question is due to Chatterjee and Varadhan (2011). Fix $0<p<r<1$ and take $G\sim G(n,p)$, the Erdős-Rényi random graph with edge density $p$, conditioned to have at least as many triangles as the typical $G(n,r)$. Is $G$ close in cut-distance to a typical $G(n,r)$? Via a beautiful new framework for large deviation principles in $G(n,p)$, Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of $(p,r)$ where the answer is positive. They further showed that for any small enough $p$ there are at least two phase transitions as $r$ varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed $d$-regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all $(p,r)$ such that $(r^d,h_p(r))$ lies on the convex minorant of $x\mapsto h_p(x^{1/d})$ where $h_p$ is the rate function of a binomial with parameter $p$. In particular, the answer for triangles involves $h_p(\sqrt{x})$ rather than the natural guess of $h_p(x^{1/3})$ where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of $G\sim G(n,p)$ is conditioned to exceed the typical value of the largest eigenvalue of $G(n,r)$. Building on the work of Chatterjee and Diaconis (2012) we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn (2001) and Galvin and Tetali (2004).
What is the probability that the number of triangles in $\mathcal{G}_{n,p}$, the Erdős-Rényi random graph with edge density $p$, is at least twice its mean? Writing it as $\exp[- r(n,p)]$, already the order of the rate function $r(n,p)$ was a longstanding open problem when $p=o(1)$, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that $r(n,p)\asymp n^2p^2 \log (1/p)$ for $p \gtrsim \frac{\log n}n$; the exact asymptotics of $r(n,p)$ remained unknown. The following variational problem can be related to this large deviation question at $p\gtrsim \frac{\log n}n$: for $δ>0$ fixed, what is the minimum asymptotic $p$-relative entropy of a weighted graph on $n$ vertices with triangle density at least $(1+δ)p^3$? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed $p$. A very recent breakthrough of Chatterjee and Dembo extended its validity to $n^{-α}\ll p \ll 1$ for an explicit $α>0$, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is $\min\{\frac12 δ^{2/3}\,,\, \frac13 δ\}$ when $n^{-1/2}\ll p \ll 1$ vs. $\frac12 δ^{2/3}$ when $n^{-1} \ll p\ll n^{-1/2}$ (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that $\mathcal{G}_{n,p}$ for $ n^{-α} \leq p \ll 1$ has twice as many triangles as its expectation is $\exp[-r(n,p)]$ where $r(n,p)\sim \frac13 n^2 p^2\log(1/p)$. Our results further extend to $k$-cliques for any fixed $k$, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when $p\geq n^{-α}$.
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 Erdős-Rényi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-δ) \mathbb{E} X_H)$ for fixed $0 < δ< 1$. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for $p \ge n^{-α_H}$ (and conjecturally for a larger range of $p$). We study this variational problem and provide a partial characterization of the so-called "replica symmetric" phase. Informally, our main result says that for every $H$, and $0 < δ< δ_H$ for some $δ_H > 0$, as $p \to 0$ slowly, the main contribution to the lower tail probability comes from Erdős-Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite $H$ and $δ$ close to 1.
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.
The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for $k$-term arithmetic progressions in pseudorandom subsets of $\mathbb{Z}_N$ of density $N^{-c_k}$. The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemerédi theorem.
The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.
We study a discrete model of repelling particles, and we show using linear programming bounds that many familiar families of error-correcting codes minimize a broad class of potential energies when compared with all other codes of the same size and block length. Examples of these universally optimal codes include Hamming, Golay, and Reed-Solomon codes, among many others, and this helps explain their robustness as the channel model varies. Universal optimality of these codes is equivalent to minimality of their binomial moments, which has been proved in many cases by Ashikhmin and Barg. We highlight connections with mathematical physics and the analogy between these results and previous work by Cohn and Kumar in the continuous setting, and we develop a framework for optimizing the linear programming bounds. Furthermore, we show that if these bounds prove a code is universally optimal, then the code remains universally optimal even if one codeword is removed.
We extend the $L^p$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence, and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs, and sparse versions of $W$-random graphs.
Tao conjectured that every dense subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This was very recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler. Here we give a simple proof using the Green-Tao theorem on linear equations in primes and the Furstenberg-Katznelson multidimensional Szemerédi theorem.
A sequence of $k$-uniform hypergraphs $H_1, H_2, \dots$ is convergent if the sequence of homomorphism densities $t(F, H_1), t(F, H_2), \dots$ converges for every $k$-uniform hypergraph $F$. For graphs, Lovász and Szegedy showed that every convergent sequence has a limit in the form of a symmetric measurable function $W \colon [0,1]^2 \to [0,1]$. For hypergraphs, analogous limits $W \colon [0,1]^{2^k-2} \to [0,1]$ were constructed by Elek and Szegedy using ultraproducts. These limits had also been studied earlier by Hoover, Aldous, and Kallenberg in the setting of exchangeable random arrays. In this paper, we give a new proof and construction of hypergraph limits. Our approach is inspired by the original approach of Lovász and Szegedy, with the key ingredient being a weak Frieze-Kannan type regularity lemma.
The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their argument and improve their bound by a constant factor using a simple geometric argument, and we extend the argument to packings in hyperbolic space, for which it gives an exponential improvement over the previously known bounds. Additionally, we show that the Cohn-Elkies linear programming bound is always at least as strong as the Kabatiansky-Levenshtein bound; this result is analogous to Rodemich's theorem in coding theory. Finally, we develop hyperbolic linear programming bounds and prove the analogue of Rodemich's theorem there as well.
Recently Conlon, Fox, and the author gave a new proof of a relative Szemerédi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemerédi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition. This note contains an alternative proof of the new relative Szemerédi theorem, where we directly transfer Szemerédi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds. The proof has three main ingredients: (1) a transference principle/dense model theorem of Green-Tao and Tao-Ziegler (with simplified proofs given later by Gowers, and independently, Reingold-Trevisan-Tulsiani-Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works), (2) a counting lemma established by Conlon, Fox, and the author, and (3) Szemerédi's theorem as a black box.
Szemerédi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erdős-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.
This is a companion note to our paper 'A relative Szemerédi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemerédi theorem for $(r+1)$-term arithmetic progressions assuming a linear forms condition. Here we show how to replace this condition with an assumption about the Gowers uniformity norm $U^r$.
We provide an upper bound to the number of graph homomorphisms from $G$ to $H$, where $H$ is a fixed graph with certain properties, and $G$ varies over all $N$-vertex, $d$-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from $G$ to $H$ when $H$ is bipartite. We also apply our techniques to graph colorings and stable set polytopes.
In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.