Independence models for integer points of polytopes

The integer points of a high-dimensional polytope P are generally difficult to count or sample uniformly. We consider a class of low-complexity random models for these points which arise from an entropy maximization problem. From these models, by way of "anti-concentration" results for sums of independent random variables, we derive general, efficiently computable upper bounds on the number of integer points of P.;We make a detailed study of contingency tables with bounded entries, which are the integer points of a transportation polytope truncated by a cuboid. We provide efficiently computable estimates for the logarithm of the number of m x n tables with specified row and column sums r1, . . . , rmc 1, . . . , cn and bounds on the entries. These estimates are asymptotic as m, n → infinitysimultaneously, given that no ri (resp., cj) is allowed to exceed a fixed multiple of the average row sum (resp., column sum).;As an application, we consider a. random, uniformly selected table with entries 2 ≤ kappa having a given sum. Responding to questions raised by Diaconis and Efron in the context of statistical significance testing, we show that the occurrence of row sums r1, . . . , rm is positively correlated with the occurrence of column sums c1, . . . , when kappa ≥ 2 and, r1, . . . , rm, c 1, . . . , cn are sufficiently extreme. We give evidence that the opposite is true for near-average values of r1, . . . , rm, c1, . . . , cn.