Spreading of Sets in Product Spaces and Hypercontraction of the Markov Operator

摘要: For a pair of random variables, $(X, Y)$ on the space $\mathscr{X} \times \mathscr{Y}$ and positive constant, $\lambda$, it is an important problem information theory to look for subsets $\mathscr{A}$ $\mathscr{X}$ $\mathscr{B}$ $\mathscr{Y}$ such that conditional probability $Y$ being in supposed $X$ larger than $\lambda$. In many typical situations order satisfy this condition, must be chosen much $\mathscr{A}$. We shall deal with most frequently investigated case when $X = (X_1,\cdots, X_n), Y (Y_1,\cdots, Y_n)$ $(X_i, Y_i)$ are independent, identically distributed pairs variables finite range. Suppose distribution all values $(x, y)$. show if above condition constant $\lambda$ goes 0, then even faster 0. Generalizations some exact estimates exponents probabilities given. Our methods reveal interesting connection so-called hypercontraction phenomenon theoretical physics.