The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarized by an exponent triple (1/2, 1/3, 2/3) representing local interface fluctuation, local roughness (or inward deviation), and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models (Alexander in Commun Math Phys 224(3): 733–781, 2001; Hammond in J Stat Phys 142(2):229–276, 2011, Commun Math Phys 310(2):455–509, 2012, Ann Probab 40(3):921–978, 2012). In this article, we offer a new perspective on this phenomenon. We consider the model of directed last passage percolation in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature, as the Ising droplets do, but now its local interface fluctuation exponent is governed by KPZ relations, and is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path’s convex hull are governed by an exponent of 3/4, and inward deviation by an exponent of 1/2. That is, the exponent triple is now (2/3, 1/2, 3/4) in place of (1/2, 1/3, 2/3). This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect, concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established by Biskup et al. (Comm Pure Appl Math 68:1483–1531, 2015), settling a conjecture of Benjamini.
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The authors thank Marek Biskup, Craig Evans and Ofer Zeitouni for useful discussions, and an anonymous referee for comments on the manuscript. R.B. was partially supported by an AMS-Simons Travel Grant during the completion of this work. S.G.’s research was supported by a Miller Research Fellowship at UC Berkeley. A.H. was supported by NSF Grant DMS-1512908.
Communicated by H. Spohn
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Basu, R., Ganguly, S. & Hammond, A. The Competition of Roughness and Curvature in Area-Constrained Polymer Models. Commun. Math. Phys. 364, 1121–1161 (2018). https://doi.org/10.1007/s00220-018-3282-x