Domain walls and chaos in the disordered SOS model

Abstract

Domain walls, optimal droplets and disorder chaos at zero temperature are studied numerically for the solid-on-solid model on a random substrate. It is shown that the ensemble of random curves represented by the domain walls obeys Schramm’s left passage formula with κ = 4 whereas their fractal dimension is ds = 1.25, and therefore their behavior cannot be described as showing ‘Schramm– (or stochastic) Loewner evolution’ (SLE). Optimal droplets with a lateral size between L and 2L have the same fractal dimension as domain walls but an energy that saturates at a value of order for such that arbitrarily large excitations exist which cost only a small amount of energy. Finally it is demonstrated that the sensitivity of the ground state to small changes of order δ in the disorder is subtle: beyond a crossover length scale Lδ∼δ−1 the correlations of the perturbed ground state with the unperturbed ground state, rescaled using the roughness, are suppressed and approach zero logarithmically.

Percolation thresholds on planar Euclidean relative-neighborhood graphs

“…describing vortices in high T c superconductivity [3,4], the negativeweight percolation problem [5,6], and domain wall excitations in disordered media such as 2D spin glasses [7,8] and the 2D solid-on-solid model [9]. Besides discrete lattice models there is also interest in studying continuum percolation models, where recent studies reported on highly precise estimates of critical properties for spatially extended, randomly oriented and possibly overlapping objects with various shapes [10].…”

Watersheds are Schramm-Loewner Evolution Curves

“…Establishing SLE for such systems has provided valuable information on the underlying symmetries and paved the way to some exact results [5,9,10]. In fact, SLE is not a general property of non-self-crossing walks since many curves have been shown not to be SLE as, for example, the interface of solid-on-solid models [11], the domain walls of bimodal spin glasses [12], and the contours of negative-weight percolation [13].Recently, the watershed (WS) of random landscapes [14][15][16], with a fractal dimension d f ≈ 1.22, was shown to be related to a family of curves appearing in different contexts such as, e.g., polymers in strongly disordered media [17], bridge percolation [14], and optimal path cracks [18]. In the present Letter, we show that this universal curve has the properties of SLE, with κ = 1.734 ± 0.005. κ < 2 is a special limit since, up to now, all known examples of SLE found in Nature and statistical physics models have 2 ≤ κ ≤ 8, corresponding to fractal dimensions d f between 1.25 and 2.…”

“…Establishing SLE for such systems has provided valuable information on the underlying symmetries and paved the way to some exact results [5,9,10]. In fact, SLE is not a general property of non-self-crossing walks since many curves have been shown not to be SLE as, for example, the interface of solid-on-solid models [11], the domain walls of bimodal spin glasses [12], and the contours of negative-weight percolation [13].…”

q-Deformed Loewner Evolution

“…On the other hand, various generalizations and modifications have been proposed by extending or changing the driving function, for instance by using Lévy flights [25], or discrete-scale invariant functions [26]. Of course, not all statistical mechanics models on the lattice naturally lead to curves whose continuum limit should be SLE (e.g., negative-weight percolation [27], disordered solid-on-solid models [28], bimodal Edwards-Anderson spin glass [29]); while conformal invariance is a natural requirement for models at criticality, the Markov property is not to be expected in general.…”