Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

• DOI: 10.1007/s00440-016-0751-z
• Publication Date: 2017-01-03T14:16:10Z
• Authors: Dmitry Ioffe; Yvan Velenik; Vitali Wachtel
• Keywords: ddc:510; Critical prewetting; Probability (math.PR); FOS: Physical sciences; Ordered random walks; Mathematical Physics (math-ph); Invariance principle; 01 natural sciences; Non-intersecting random walks; Non-crossing random walks; 60G50, 60F17; 0103 physical sciences; FOS: Mathematics; ddc:510; 0101 mathematics; info:eu-repo/classification/ddc/510; Entropic repulsion; Mathematics - Probability; Mathematical Physics; Ferrari-Spohn diffusions

We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of $2+1$-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari--Spohn diffusions associated with limiting Sturm--Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.<br/>33 pages<br/>