Area distribution and the average shape of a Lévy bridge

Abstract

We consider a one dimensional Lévy bridge x_B of length n and index 0 < α< 2, i.e. a Lévy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a Lévy bridge and show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/α} F_α(A/n^{1+1/α}), with the asymptotic behavior F_α(Y) \sim Y^{-2(1+α)} for large Y. For α=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions. We also compute the average profile < \tilde x_B (m) > at time m of a Lévy bridge with fixed area A. For large n and large m and A, one finds the scaling form < \tilde x_B(m) > = n^{1/α} H_α({m}/{n},{A}/{n^{1+1/α}}), where at variance with Brownian bridge, H_α(X,Y) is a non trivial function of the rescaled time m/n and rescaled area Y = A/n^{1+1/α}. Our analytical results are verified by numerical simulations.

21 pages, 4 Figures

References in corpus (16)

• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• Interaction driven real-space condensation
• On the area under a continuous time Brownian motion till its first-passage time
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics
• From urn models to zero-range processes: statics and dynamics
• Tuning the shape of the condensate in spontaneous symmetry breaking
• Mass condensation in one dimension with pair-factorized steady states
• Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution
• Multiscaling in the sequence of areas enclosed by coalescing random walkers
• Area Distribution of Elastic Brownian Motion

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