Universality and crossover behavior of single-step growth models in 1+1 and 2+1 dimensions

The hidden fluctuation-dissipation theorem for growth (a)

“…In order to model the EW equation we shall use the SS model. The SS model has gained a lot of importance over the years [38][39][40][41][42]: firstly, because it was proved to be a KPZ model and secondly due to its connection with other models, such as the asymmetric simple exclusion process [40], the six-vertex model [41,43,44], and the kinetic Ising model [41,45]. Finally, our interest here is because this model, in particular, can become an EW model.…”

“…Note that for p = q, it becomes the EW model. Changing the probabilities we can get a lot of relevant information [42]. Now, we start to simulate the model with p = q = 1/2, i.e.…”

The Fractal Geometry of Growth: Fluctuation–Dissipation Theorem and Hidden Symmetry

“…Here, the average is taken over the space. The roughness is a very important physical quantity since many important phenomena have been associated with it [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For many growth processes, the roughness, w(L,t), increases with time until reaches a saturated roughness w s , i.e., w(t →∞) w s .…”

“…As a consequence, most of these stochastic systems are interconnected. For instance, the SS model [10][11][12][13][14], which is connected with the asymmetric simple exclusion process [12], the six-vertex model [13,20,21], and the kinetic Ising model [13,22], all of them are of fundamental importance. It is noteworthy that quantum versions of the KPZ equation have been recently reported that are connected with a Coulomb gas [23], a quantum entanglement growth dynamics with random time and space [24], as well as in infinite temperature spin-spin correlation in the isotropic quantum Heisenberg spin-1/2 model [25]; [26].…”

“…Note that the symmetry h → −h is violated for KPZ due to the presence of the non-linear dependence on the local slope [ ∇h] 2 . The KPZ equation describes very well the dynamics of some atomistic models such as the etching model [4][5][6][7][8][9], and the Single-Step (SS) model [10][11][12][13][14] in the long wavelength limit. For atomistic models we define our Euclidean space as a d-dimensional hypercubic lattice within the region Ω ⊂ R d , with volume V = L d , where L is the lateral side.…”