A Relation between Brownian Bridge and Brownian Excursion

Scaling limits of equilibrium wetting models in (1+1)–dimension

“…To summarize, we have: P 0,N • (X N ) −1 converges weakly in C([0, 1]) to the law of β; by (21) Q c 0,N is the law of E under P 0,N • (X N ) −1 ; by [18], e = E(β) in distribution and P(β ∈ D E ) = 0. By the Mapping Theorem (see e.g.…”

“…In Section 8 we complete the proof of convergence at criticality, by reconstructing the full trajectories starting from the zero level sets. We point out that a central quantity in our analysis is the partition function associated to the constrained case and some of the arguments concentrate first on that case: when this strategy is possible the result goes over to the free case, as one would expect, without much troubles, essentially thanks to formula (18). This is however not always possible: in particular for = c , the estimates in the constrained case are more delicate and involve careful work with Radon-Nykodym derivatives for variables at sites away from the boundary: see section 7.…”

“…where in the denominator of the last term we use (18). By (16) and Lemma 3 we can replace Z c δ,t P (N − t) by t −3/2 (N + 1 − t) −1/2 times positive constants both as upper and lower bounds.…”

“…We define now measurable maps ζ : We recall now that, by [18], e = E(β) in distribution. Since a.s. e > 0 on (0, 1) and e(0) = 0, we have that a.s. ζ(β) is the unique time in [0, 1] where β attains its minimum.…”

Fluctuations for a ∇φ interface model with repulsion from a wall

“…Therefore, (15) follows from (13)-(14) for V = V 0 and from Lemma 2. However, in the general case V = V 0 our proof of (13) is not direct, but follows from (14), (15) and the convergence of the l.h.s. of (12).…”