A figure of merit is defined for an indexing of a given pattern as M20 = Q2o/2gN2o. Here N_,0 is the number of different calculated Q values up to Q20, which is the Q value for the 20th observed and indexed line;is the average discrepancy in Q for these 20 lines. From a number of indexings which have been disproved by single-crystal analysis, the conclusion is drawn that M20 < 6 must give rise to considerable doubt about the result. A number of confirmed indexings shows values of 20-60 for good routine work on pure, well-crystallized samples, and down to 6 for retrievable correct indexings of less accurate data. If the number of unindexed lines below Q20 is not more than two, a value M:0 > 10 guarantees that the indexing is substantially correct.

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalitiesSuch Sobolev type inequalities hold, e.g., if the underlying measure satisfies the logSobolev inequality (in which case C( p) ≤ C √ p) or the Poincaré inequality (then C( p) ≤ C p). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of f . When the underlying measure is Gaussian and f is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.

Abstract. An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on Hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest.