PHYSICAL RYan et al. Reply: In his Comment [1] Kim calculates the quantity Ar and claims that a "sudden jump" in Av is evidence that the phase transition that we found [2,3] is trivial in 2-hi dimensions, much like the transition of Amar and Family [4] where the coupling X, in the Kardar-Parisi-Zhang equation, changes sign as a function of control parameter [5]. Ar is the average over the surface of X(p)\\h\ 2 .Kim is correct that this function varies rapidly. The corresponding X, if we assume that (|V/j| 2 > is a constant of order unity, is small for /?=^0.3, where we locate the transition. Thus we might be in a crossover regime from the X =0 fixed point. This is Kim's argument as we understand it.We first note that we do not think that Ar is the best quantity for assessing our model. We will focus in what follows on X, as is conventional in this field. The ratio between Ar and A., (|V7?| 2 >, is not a constant and depends on p. We show this by calculating X itself [5] by forcing an average tilt 0 on the surface and setting X=jd 2 v/ d0 2 |0=o-We present our results in Fig. 1 plotted together with Kim's results. Kim is correct that X(p) is nonlinear; in our paper we were wrong on this point. However, an examination of the figure will show that X is smooth, has no sudden jump, and is fairly large in the important region 0.2 for 0.2
is small for small p because the surface is smoother.Of course, the fact that X(p) is nonlinear is not a problem, in principle. It is sufficient for our purposes that X(p) is continuous, monotonic, and zero only where p =0, just as a nonlinear thermometer can indicate an ordinary thermal transition. X certainly does not change sign as in Ref.[4]. We correctly found the (noncontroversial) transition in 34-1 dimensions using the same thermometer and the absence of a transition in 1 -I-1 dimensions.There remains only a numerical question: Is A,(0.2) ^ 0.12 so small that we could not distinguish it from 0? Kim's suggestion is that X =0.12 appears to give weak coupling because our sample sizes are too small and we have a finite-size crossover. To answer this sort of criticism, the proper tool is finite-size scaling. This is exactly what we did [2] in our paper. In 3+ 1 dimensions we got expected and reasonable results. In 2+1 dimensions, below the putative transition, the effective roughness exponent a became smaller for large systems. In the absence of a phase transition this is very hard to understand in finite-size scaling, which says that a should cross over to larger values. The behavior that we found is clearly not a simple crossover as Kim implies and has been suggested [6] elsewhere. It may be a phase transition or a more complicated type of crossover [3j, but that is another issue, not addressed by Kim's Comment.
