($2\mathbf{+}1$)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions

We examine numerically the zero-temperature ($2+1$)-dimensional directed polymer in a random medium, along with several of its brethren via the Kardar-Parisi-Zhang (KPZ) equation. Using finite-size and KPZ scaling Ansätze, we extract the universal distributions controlling fluctuation phenomena in this canonical model of nonequilibrium statistical mechanics. Specifically, we study point-point, point-line, and point-plane directed polymer geometries, scenarios which yield higher-dimensional analogs of the Tracy-Widom distributions of random matrix theory. Our analysis represents a robust, multifaceted numerical characterization of the $2+1$ KPZ universality class and its limit distributions.

Figure 1

Fluctuation PDFs: $2+1$ KPZ models.

Figure 2

Disorder-averaged, parabolic DPRM free energy profile. Insets: (a) Summary Krug-Meakin plot for $2+1$ KPZ class models, (b) quadratic KPZ tilt-dependent growth velocity [23] for the $2+1$ RSOS model.

Figure 3

Universal limit distribution, $2+1$ KPZ class: DPRM point-plane geometry; KPZ stochastic growth with flat interface IC.

Figure 4

Universal PDFs: $2+1$ DPRM point-point and point-line geometries. Table inset: Distribution moments.