Recovery, detection and confidence sets of communities in a sparse stochastic block model

Posterior distributions for community assignment in the planted bi-section model are shown to achieve frequentist exact recovery and detection under sharp lower bounds on sparsity. Assuming posterior recovery (or detection), one may interpret credible sets (or enlarged credible sets) as consistent confidence sets. If credible levels grow to one quickly enough, credible sets can be interpreted as frequentist confidence sets without conditions on the parameters. In the regime where within-class and between-class edge-probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage. The diameters of credible sets are controlled and match rates of posterior convergence.Comment: 22 pp., 2 fi

Effective equidistribution of primitive rational points on expanding horospheres

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems --- an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical subgroup in any dimension. We apply our result to provide a rate of convergence to the limiting distribution for the appropriately rescaled diameters of random circulant graphs.Comment: 21 pages, incorporates the referee's comments and correction

On the dimension growth of groups

Dimension growth functions of groups have been introduced by Gromov in 1999. We prove that every solvable finitely generated subgroups of the R. Thompson group $F$ has polynomial dimension growth while the group $F$ itself, and some solvable groups of class 3 have exponential dimension growth with exponential control. We describe connections between dimension growth, expansion properties of finite graphs and the Ramsey theory.Comment: 20 pages; v3: Erratum and addendum included as Section 9. We can only prove that the lower bound of the dimension growth of $F$ is exp sqrt(n). New open questions and comments are added. v4: The paper is completely revised. Dimension growth with control is introduced, connections with graph expansion and Ramsey theory are include