This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class-an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

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 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 ℓ/m goes to some positive limit β < 1, the size B m,ℓ

Abstract. The Airy distribution (of the "area" type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders −1, −3, −5, etc., as well as + 1 3 , − 5 3 , − 11 3 , etc. and − 7 3 , − 13 3 , − 19 3 , etc. Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on "non-probabilistic" arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +∞, and power symmetric functions of the zeros −α k of Ai(z).Key Words. Brownian excursion area, Airy function, Parking problem, Linear probing hashing.Introduction. For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The name is derived from the connection between Brownian motion and the Airy function, a fact discovered around 1980 by several authors; see [16] and [20]. For combinatorialists and theoretical computer scientists, this Airy distribution (of the "area type") arises in a surprising diversity of contexts like parking allocations, hashing tables, trees, discrete random walks, mergesorting, etc.The most straightforward description of the Airy distribution is by its moments themselves defined by a simple nonlinear recurrence. We follow here the notations and the normalization of [11].

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