Phase transition for parking blocks, Brownian excursion and coalescence

Abstract

In this paper, we consider hashing with linear probing for a hashing table with m places, n items (n < m), and l = m<n empty places. For a non computer science-minded reader, we shall use the metaphore of n cars parking on m places: each car chooses a place at random, and if this place k is occupied, the car tries successively k+1, k+2, ... until it finds an empty place (with the convention that place m+1 is actually place 1). Pittel [42] proves that when l/m goes to some positive limit a < 1, the size of the largest block of consecutive cars is O(log m). In this paper we examine at which level for n a phase transition occurs for the largest block of consecutive cars between o(m) and O(m). The intermediate case reveals an interesting behaviour 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.