Abstract: 

(i) We study the statistics of stretched two-dimensional random walks (Brownian bridges) in the vicinity of an impermeable disc using the optimal–fluctuation approach. We show that the transverse span of the walks away from the boundary scales with the Kardar–Parisi–Zhang (KPZ) exponent $1/3$. (ii) Using the analogy between the optimal fluctuation in this setting and that in the one-dimensional Balagurov–Vaks trapping problem, we propose a connection between KPZ-like statistics and Lifshitz tails arising in a deterministic large-deviation landscape. (iii) By interpreting the radial component of the random walk above the disc as diffusion in a conformally invariant $1/r^{2}$ potential, we recover the Efimov–BKT behavior of the associated renormalization-group flow. From a large-deviation perspective, we argue that the typical paths responsible for the BKT-like behavior belong to a sub-ensemble of stretched Brownian bridges driven into the large-deviation regime.