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Abstract
We consider in this paper the support $[L',R']$ of the one dimensional Integrated Super Brownian Excursion. We determine the distribution of $(R',L')$ through a modified Laplace transform. Then we give an explicit value for the first two moments of $R'$ as well as the covariance of $R'$ and ${L'}$.
Highlights
Introduction and resultsThe motivation of this work comes from the paper of Chassaing and Schaeffer [2]
They prove the rescaled radius of random quadrangulation converges, as the number of faces goes to infinity, to the width r = R − L of the one dimensional Integrated Super Brownian Excursion (ISE) support [L, R ]
It is of particular interest to compute the law of r = R − L as well as its moments
Summary
The motivation of this work comes from the paper of Chassaing and Schaeffer [2] They prove the rescaled radius of random quadrangulation converges (in law), as the number of faces goes to infinity, to the width r = R − L of the one dimensional Integrated Super Brownian Excursion (ISE) support [L , R ] (see [1] and the reference therein for the definition of the ISE). We give the following numerical approximation (up to 10−3): E[R ] 2.580, Var(R ) 0.863 and for r = R − L , E[r ] 5.160 and Var(r ) 0.651 To prove those results, we use the fact that the ISE has the same distribution (up to a constant scaling) as the total mass of an excursion of the Brownian snake conditioned to have a duration σ of length 1.
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