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Let M n M_n be drawn uniformly from all ± 1 \pm 1 symmetric n × n n \times n matrices. We show that the probability that M n M_n is singular is at most exp ( − c ( n log n ) 1 / 2 ) \exp (-c(n\log n)^{1/2}) , which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of exp ( − c n 1 / 2 ) \exp (-c n^{1/2}) on the singularity probability, our method is different and considerably simpler: we prove a “rough” inverse Littlewood-Offord theorem by a simple combinatorial iteration.more » « less
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Abstract We determine the asymptotics of the number of independent sets of size $$\lfloor \beta 2^{d-1} \rfloor$$ in the discrete hypercube $$Q_d = \{0,1\}^d$$ for any fixed $$\beta \in (0,1)$$ as $$d \to \infty$$ , extending a result of Galvin for $$\beta \in (1-1/\sqrt{2},1)$$ . Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $$Q_d$$ drawn according to the hard-core model at any fixed fugacity $$\lambda>0$$ . In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.more » « less
