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View article: A Bayesian Proof of the Spread Lemma
A Bayesian Proof of the Spread Lemma Open
A key set‐theoretic “spread” lemma has been central to two recent celebrated results in combinatorics: the recent improvements on the sunflower conjecture by Alweiss, Lovett, Wu, and Zhang; and the proof of the fractional Kahn–Kalai conjec…
View article: Sharp thresholds in inference of planted subgraphs
Sharp thresholds in inference of planted subgraphs Open
A major question in the study of the Erdős--Rényi random graph is to understand the probability that it contains a given subgraph. This study originated in classical work of Erdős and Rényi (1960). More recent work studies this question bo…
View article: Locality of critical percolation on expanding graph sequences
Locality of critical percolation on expanding graph sequences Open
We study the locality of critical percolation on finite graphs: let $G_n$ be a sequence of finite graphs, converging locally weakly to a (random, rooted) infinite graph $G$. Consider Bernoulli edge percolation: does the critical probabilit…
View article: A second moment proof of the spread lemma
A second moment proof of the spread lemma Open
This note concerns a well-known result which we term the ``spread lemma,'' which establishes the existence (with high probability) of a desired structure in a random set. The spread lemma was central to two recent celebrated results: (a) t…
View article: On the Second Kahn--Kalai Conjecture
On the Second Kahn--Kalai Conjecture Open
For any given graph $H$, we are interested in $p_\mathrm{crit}(H)$, the minimal $p$ such that the Erdős-Rényi graph $G(n,p)$ contains a copy of $H$ with probability at least $1/2$. Kahn and Kalai (2007) conjectured that $p_\mathrm{crit}(H)…
View article: Sharp threshold sequence and universality for Ising perceptron models
Sharp threshold sequence and universality for Ising perceptron models Open
We study a family of Ising perceptron models with $\{0,1\}$-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show th…
View article: Gardner formula for Ising perceptron models at small densities
Gardner formula for Ising perceptron models at small densities Open
We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2…
View article: Capacity lower bound for the Ising perceptron
Capacity lower bound for the Ising perceptron Open
We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube {−1,+1}N intersected by M random half-spaces. The perceptron's capacity is the largest integer MN for which the intersection is nonempty. It …
View article: Breaking of 1RSB in random MAX-NAE-SAT
Breaking of 1RSB in random MAX-NAE-SAT Open
For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satisfiability transition occurs. For random $k$-SAT and related models it happens at clause density $α$ around $…
View article: Capacity lower bound for the Ising perceptron
Capacity lower bound for the Ising perceptron Open
We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube $\{-1,+1\}^N$ intersected by $M$ random half-spaces. The perceptron's capacity is $α_N \equiv M_N/N$ for the largest integer $M_N$ such that …
View article: Spectral algorithms for tensor completion
Spectral algorithms for tensor completion Open
In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational…
View article: The number of solutions for random regular NAE-SAT
The number of solutions for random regular NAE-SAT Open
Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirmin…
View article: Maximum independent sets on random regular graphs
Maximum independent sets on random regular graphs Open
We determine the asymptotics of the independence number of the random d-regular graph for all ${d\\geq d_0}$ . It is highly concentrated, with constant-order fluctuations around ${n\\alpha_*-c_*\\log n}$ for explicit constants ${\\alpha_*(…
View article: Shotgun assembly of random regular graphs
Shotgun assembly of random regular graphs Open
Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider…
View article: Supercritical minimum mean-weight cycles
Supercritical minimum mean-weight cycles Open
We study the weight and length of the minimum mean-weight cycle in the stochastic mean-field distance model, i.e., in the complete graph on $n$ vertices with edges weighted by independent exponential random variables. Mathieu and Wilson sh…