2003/08/31 by Patrik L. Ferrari, Herbert Spohn · 3 citations
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60J60 #msc:60J65
paper · pdf · doi:10.1214/009117905000000125
published as Annals of Probability 2005, Vol. 33, No. 4, 1302-1325 · Published at http://dx.doi.org/10.1214/009117905000000125 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2005/07/01 · arxiv created 2005/08/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T)=0 conditioned to stay above the semicircle cT(t)=√T2-t2. In the limit of large T, the fluctuation scale of b(t)−cT(t) is T1/3 and its time-correlation scale is T2/3. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t=τT, τ∈(−1,1), is only through the second derivative of cT(t) at t=τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height Tγ, γ>1/2. The fluctuation scale is then T(2−γ)/3. More general conditioning shapes are briefly discussed.