Equilibrium Restricted Solid-on-Solid Models with Constraints on the Growth of Extremal heights Yup Kim, Sooyeon Yoon Kyunghee University Hyunggyu Park Inha University

Abstract even-visited random walks. 1 Dynamical scaling properties of an equilibrium restricted solid-on-solid (RSOS) model with a constraint on the growth processes at extremal heights are investigated by simulations, where the equilibrium means the case in which evaporation growth process occurs with the same probability as deposition growth process. Except for the RSOS condition we imposed a constraint on the growth processes which make new maximal or minimal height of the surface is accepted with a probability p. In d=1 the surface fluctuation W in the saturation regime (or t >> Lz) satisfies W = La (a=1/3) for p < 1. In d=2 the dynamic exponent z is found to z  2.5. These results are physically explained by the relation of the present so-called the (stochastic) even-visited random walks.

2 Model Step 1. Find the global maximum (minimum) height hmax (hmin) among the height distribution in a d–dimensional hypercubic substrate. Step 2. Select randomly a site . Step 3. a) If or , Deposition is attempted with the probability . Evaporation is attempted with the probability 1- . b) If or , Deposition is attempted with the new probability p. Evaporation is attempted with the same probability p.

3 * All accepted site should be satisfied with the restricted solid-on-solid (RSOS) constraint, ( is one of nearest-neighbor bond vectors of a site in d-dimensional hypercubic lattice.) p () p (1-)  1-

Physical Backgrounds for This Study 4 Physical Backgrounds for This Study Steady state or Saturation regime, 1. Simple RSOS with Normal Random Walk(1d) 2. Two-site correlated growth (Yup Kim, T. Kim and H. Park, 2002) Dimer growth (J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE, 2001, J. D. Noh, H. Park and M. den Nijs, PRL, 2000)  =-1,  =1, nh=even number, Even-Visiting Random Walk (1d)

What happens at  = 0,  = 1 (1d) ? 5 What happens at  = 0,  = 1 (1d) ? New hmax, hmin p ( PNew({hr}RSOS)=1/2) : Acceptance Probability of the New hmax, hmin

Even-Visiting Random Walk 6 At  = 0,   1 (1d), Phase Diagram  = 0  = 1  =-1  =1 Normal Random Walk Even-Visiting Random Walk ?

Simulation Results  1d p eff 0.1 0.33(1) 0.3 0.6 0.32(1) 0.8 0.344 7 512, 1024, 2048

8  = 0.22 L = 1024  = 0.22(1)

Scaling Collapse to in 1d. ( = 1/3 , z = 1.5) 9 Scaling Collapse to in 1d. ( = 1/3 , z = 1.5)

(Deok-Sun Lee and Marcel den Nijs, 10  2d ? (Deok-Sun Lee and Marcel den Nijs, cond-mat, 0110485, 2001) Slope of Normal RSOS p Slope a N. RSOS(1) 0.174 0.1 0.5

11 Model Slope a N. RSOS 0.174 p = 0.5 Dimer growth 0.16 Two-site growth

From W2(t<<Lz), we can estimate z as z  2.5. 12 Scaling Theory From W2(t<<Lz), we can estimate z as z  2.5.

Scaling Collapse in 2d. ( z  2.5) 13 Scaling Collapse in 2d. ( z  2.5)  

Conclusion Yes ! ? ??? Is that Really Right?????  1d  =1  = 0  = 1 14 Conclusion  1d Yes !  =1  =-1  = 0  = 1 ? Even-Visiting Random Walk Normal Random Walk Is that Really Right?????  2d ???  =1  =-1  = 1  = 0 Even-Visiting Random Walk Normal Random Membrane

  > 1/2 (growing phase),  < 1/2 (eroding phase) 15   > 1/2 (growing phase),  < 1/2 (eroding phase)  Normal RSOS Model  Surface Dynamics in t << Lz regime  What is the Corresponding Continuum Equation ?