DescriptionThis thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. Random matrices: We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to infinity as n does. Our result complements the McKay law [19], which applied for the case d is an absolute constant. Random boxes. Take n random boxes with axis-parallel edges inside the unit cube [0; 1][superscript]d, the piercing number is the minimum number of points needed to pierce all boxes. Using hypergraph setting, we was able to prove a near sharp estimation for the piercing number. This thesis is based on two papers by the author [31] and [30] (joint work with Van Vu and Ke Wang).