Clusters crossed by a random walk in two-dimensional systems

How many distinct neighborhoods could a walking pedestrian encounter in a city? The naive "area law" expectation suggests a linear scaling of the number of features as a function of path length, so the walker sees more neighborhoods (clusters) as he walks a longer distance. However, interesting phenomena occur when the underlying structure is critical, for sharp turns in the walker's path lead to a universal logarithmic decrease in the cluster count [1]. This observation can be used to detect if the underlying clusters are critical and pinpoint the universality class [2]. Here we consider a random walk with constant, unpredictable turns, leading to an unexplored singular limit that potentially reveals more information about the underlying complex system. Due to the area law, multiplicative poly-logarithmic corrections exist even if the cluster structure is off-critical [3]. We hypothesize a combination of both multiplicative and additive logarithmic corrections for critical clusters and confront this assumption both numerically and analytically.
[1] Kovacs I.A. et al. PRB 86(2012)214203
[2] Kovacs I.A. et al. PRB 89(2014)064421
[3] van Wigland F. et al. J. Phys. A: Math. Gen 30(1996) 507-531