Phase transition for Parking blocks,Brownian excursion and coalescence

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n < m), and ? = m ? n empty places. For a noncomputer science‐minded reader, we shall use the metaphore of n cars parking on m places: each car c_i chooses a place p_i at random, and if p_i is occupied, c_i tries successively p_i + 1, p_i + 2, until it finds an empty place. Pittel 42] proves that when ?/m goes to some positive limit β < 1, the size B of the largest block of consecutive cars satisfies 2(β ? 1 ? log β)B = 2 log m ? 3 log log m + Ξ_m, where Ξ_m converges weakly to an extreme‐value distribution. In this paper we examine at which level for n a phase transition occurs between B = o(m) and m ? B = o(m). The intermediate case reveals an interesting behavior of sizes of blocks, related to the standard additive coalescent in the same way as the sizes of connected components of the random graph are related to the multiplicative coalescent. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 76–119, 2002