Iterative Schwarz-Christoffel Transformations
Driven by Random Walks and Fractal Curves
Stochastic Loewner evolution (SLE) is a differential equation driven by a one-dimensional Brownian motion (BM), whose solution gives a stochastic process of conformal transformation on the upper half complex-plane . As an evolutionary boundary of image of the transformation, a random curve (the SLE curve) is generated, which is starting from the origin and running in toward the infinity as time is going. The SLE curves provides a variety of statistical ensembles of important fractal curves, if we change the diffusion constant of the driving BM. In the present paper, we consider the Schwarz-Christoffel transformation (SCT), which is a conformal map from to the region with a slit starting from the origin. We prepare a binomial system of SCTs, one of which generates a slit in with an angle from the positive direction of the real axis, and the other of which with an angle . One parameter is introduced to control the value of and the length of slit. Driven by a one-dimensional random walk, which is a binomial stochastic process, a random iteration of SCTs is performed. By interpolating tips of slits by straight lines, we have a random path in , which we call an Iterative SCT (ISCT) path. It is well-known that, as the number of steps of random walk goes infinity, each path of random walk divided by converges to a Brownian curve. Then we expect that the ISCT paths divided by (the rescaled ISCT paths) converge to the SLE curves in . Our numerical study implies that, for sufficiently large , the rescaled ISCT paths will have the same statistical properties as the SLE curves have, supporting our expectation.
pacs:05.40.-a, 05.45.Df, 02.30.-f
One of the highest topics of recent progress in statistical physics of critical phenomena and random fractal patterns is introduction of the Stochastic Loewner Evolution (SLE) by Schramm Sch00 (); Law05 (); Law07 (). The SLE will provide a unified theory of statistics of random curves in the plane, which covers (continuum limits of) random interfaces characterizing surface critical phenomena in equilibrium (e.g. the percolation exploration process, the critical Ising interface), models of random chains in polymer physics (e.g. the self-avoiding walks), fractal patterns playing important roles in non-equilibrium statistical mechanics models (e.g. the loop-erased random walks and the uniform spanning trees for sandpile models and forest fire models showing self-organized criticality), and so on KN04 (); Car05 (); BB06 (). The theory is based on two branches of mathematics, the complex function theory Pom75 () and the stochastic analysis Law05 (); Law07 () and is strongly connected with the conformal field theory Car01 (); BB03 (); FW03 ().
As well as by wideness of applications and by richness of mathematics, we are attracted by simple setting of the theory ; Consider a complex plane . (i) First we consider a motion of Brownian particle on the real axis . We assume that it starts from the origin 0 at time and the diffusion constant is given by . If we write the position of the diffusion particle on at time as , then and for . Usually we denote the position of a one-dimensional standard Brownian motion (BM) at time by , for which , . The BM has the scaling property such that for any constant , the distribution of the position of BM at time is equal to that of the position of BM at time multiplied by a factor , that is, the equality holds in distribution. Then we can give the above by
(ii) Then we solve the following partial differential equation for a complex function on the upper half complex-plane ,
under the initial condition . (iii) Note that the boundary of consists of the real axis and an infinity point. When approaches the special point on , the position of the BM, the RHS of (2) diverges. If we trace the image of this singular point
we will have a curve in starting from the origin. For each the curve and the region enclosed by parts of the curve and the real axis should be eliminated from in order to continue to solve Eq.(2) for .
For any deterministic simple curve in , Loewner proved that there exists a real-valued function and a conformal map , which is one-to-one from to , the upper half complex-plane with a slit , such that and solve Eq.(2) with the condition (3). The equation (2) is called the Loewner equation Pom75 (); SLEa (). Schramm considered an inverse problem: given on and derive a curve by solving Eq.(2). Since he gave by a BM as Eq.(1), given by Eq.(3) performs a stochastic motion and the obtained curve is statistically distributed in Sch00 (). Equation (2) driven by a BM with variance is called the stochastic Loewner equation or the Schramm-Loewner evolution (SLE) SLEb (), and a random curve is called the SLE curve with parameter (the SLE curve).
Although the change of diffusion constant causes only quantitative change of the driving function , that is, scale/time change in distribution, it does qualitative change of SLE curves. When , the curve is simple (with no self-intersection) with . When , the curve is self-intersecting and hits the real axis infinitely many times, but it is not dense on : with . And when , it will cover whole of in Law05 (); Law07 (). The fractal dimension (Hausdorff dimension) of the SLE curve is determined as Bef08 (); Law09 ()
Moreover, if is chosen to be a special value, SLE curves provide the statistical ensembles of continuous limits of random discrete paths studied in statistical mechanics models exhibiting critical phenomena or in fractal models on lattices. For example, the value is for the critical percolation model Smir01 (); Bef07 (). Effective methods for numerical simulations of SLE curves by computers have been reported Ken07 (); Ken09 (); KMN09 ().
Today we can learn from mathematics literatures that the Loewner equation (2) for a deterministic function had played important roles in the complex function theory even before Schramm introduced its stochastic version Pom75 (). We shall say, however, that this equation has not been familiar to us, physicists. More familiar differential equation to us in the complex calculus is the one, whose solution gives a conformal transformation from to the interior of a polygon on the complex plane with mapping the real axis to a piecewise linear boundary of the polygon, called the Schwarz-Christoffel transformation (SCT) (see, for example, AF03 ()). So here we try to discuss the Loewner equation (2) by using a special case of SCT.
Let . Consider a conformal map from to the upper half complex-plane with a straight slit starting from the origin, , where the angle between the slit and the positive direction of the real axis is as shown in Fig.1. Since with the straight slit can be regarded as a polygon with the interior angles on the left side of the origin, around the tip of the slit, and on the right side of the origin, for any length of a slit, the conformal map is given as an SCT, which solves the differential equation
where is a complex number and are real numbers satisfying the inequalities, . By this transformation, both of the points and on are mapped to the origin, , and to the tip of the slit. See Fig.1. We have found the general solution of (5) expressed by
where is a complex number, (the gamma function), and is Gauss’s hypergeometric function . We impose the hydrodynamic condition
Then and , and we have . Eq.(5) is rewritten in this case as
We then introduce a parameter “time” and assume that and , and thus also , depend on by setting
The differential of with respect to is written as
with . Let with constants and . Then we can see that, if and only if and , becomes independent both of and ; . Combining this observation with Eq.(8) gives the following result: For , the SCT
As time goes, the straight slit performs as an “evolutionary boundary” of the image of by , in which the tip of the slit (3) is evolving as
One can observe the equality
since . That is, the driving function for the above SCT (11) is the positive or the negative root square of the squared average of the random driving function (1) of the SLE. So if we are able to introduce fluctuations into the SCT systematically, we can draw approximations of SLE curves on . It may be the basic idea to simulate SLE curves by dividing a time period into small intervals by setting and the above single SCT is replaced by an -multiplicative map of infinitesimal SCTs with sufficiently large Ken07 (); Ken09 ().
On the other hand, we know the fact that the diffusion property of BM can be observed in long-time asymptotic behavior of a simple discrete-time stochastic process, random walk (RW). Let be independent and identically distributed (i.i.d.) random variables taking values with probability 1/2 and with probability 1/2. Consider a simple symmetric RW on the one-dimensional lattice starting from the origin 0 at time . We denote the position of the random walker at time by . Then, and
In the present paper, we consider an SCT as a functional of a random variable and consider an Iterative system of SCTs (ISCTs) driven by RW. By this system, each time series of steps of RW is mapped to a series of points in . Let and be the interpolations by straight lines of and , respectively. Since for , converges to as in distribution,
which we call the rescaled ISCT path with up to time , will converge to an SLE curve up to time , , in distribution as . Figure 2 shows the rescaled ISCT paths with for and 6, which are drawn by interpolating the series of points by lines. They seems to approximate the SLE curves very well. In the present paper, we will show that, for sufficiently large , has the same statistical properties as the SLE curve has.
The paper is organized as follows. In Sec.II, we define the ISCT driven by random walk with a parameter defined by (12) and the ISCT paths. In Sec.III, by observing the behavior of ISCT paths for small values of , we show that, even if we use the same realization of RW as a driving function, the ISCT paths with large values of exhibit much more complicated motion on than those with smaller values of . In Sec.IV, we report the properties of the rescaled ISCT paths with based on large scaled computer simulations with . We evaluate the fractal dimensions of the limits of for several values of , and the dependence of on is compared with that of the fractal dimensions of the SLE curves given by (4). We also show that, for , the generalized version of Cardy’s formula of SLE curves Car92 (); Law05 () will be applicable to the rescaled ISCT paths , if and are sufficiently large. Section V is devoted to give concluding remarks. Appendix A is prepared for giving recurrence relations, which will be useful to analyze the ISCT paths.
Ii ISCT driven by RW
Noting that Eq.(12) is solved for as , we set
for , and define the SCT as a functional of a random variable by
Given one step of RW on , we consider an SCT, , which is a conformal map from to the region with a straight slit. The straight slit starts from the origin and ends at the tip located at
Next assume that two steps of RW, , is given. We transform by an SCT, . The image of , , is the region with the straight slit, which starts from the origin and ends at . Then we consider the transformation of the region with this straight slit by another SCT, . By this SCT, a straight slit from the origin to the point is generated as shown by (19). The image by of the straight slit between the origin and in is, however, no longer a straight line but a curvy one. It starts from and ends at
We have then a set of two points in . Set and now we assume that a realization of RW on up to time is specified by the series of steps, . Let
for and . We perform the following iteration of SCTs,
Then we have a curve consisting of a straight slit between the origin and and segments of curvy slits, which are sequentially connected at , and the tip is at . See Fig. 3. In the present paper we call the iterative Schwarz-Christoffel transformation (ISCT for short) driven by RW specified by .
For , we define the points in by
We set . The sequence of points is interpolated by straight lines. We call it an ISCT path in and denote it by . In other words, each realization of RW on is mapped to a path on by the ISCT.
For a given , we introduce the following recurrence relations for a series ,
with . Since are i.i.d., the recurrence formula (24) is useful to calculate the position , which is given by . Moreover, if we introduce a parameter with the relation
(17) is written as
and (19) is given by
Note that , where indicates complex conjugate. The recurrence relation (24) is then written as
for with . The expressions for the relations between two real components of complex variable and those of are given in Appendix A.
Iii Networks on generated by RW paths
For a fixed time period , consider a collection of all realization of steps of RW on ,
The total number of realizations of RWs is . We note that each realization of RW up to time is represented by a directed path on a squared lattice in a triangular region in the spatio-temporal plane,
Here any edge connecting nearest neighbor vertices in is assumed to be directed in the positive direction of the time axis; , and each path is a sequence of edges all in the positive direction, starting from to with . Figure 4 shows and an example of a realization of RW with .
By collecting all ISCT paths up to time , , we have a network on ,
Figure 5 shows for and 8 for . As the parameter increases, the network becomes spreading wider in . There the ISCT path corresponding to the realization of RW shown in Fig. 4 is indicated by a bold line for each value of . In Fig. 6, we compare the ISCT paths with (b) and (c) both obtained from the same realization (a) of RW with steps, where the networks up to , , are also shown in the background for each . We can see that the ISCT path with is much more complicated than the path with . As shown by Fig.5 and Fig.6, the network is bounded by the rightmost path generated by and the leftmost path generated by . The height of and , , is observed to converge to a positive constant in . The numerical values are given by , 0.010 (), 0.0084 (), 0.0074 (), 0.0067 (), 0.0062 (), 0.0058 (), and 0.0055 ().
Iv Numerical Analysis of curves generated by ISCT
For a given number of steps of RW, we have defined the rescaled ISCT path by Eq.(16). In particular, is obtained by interpolating the series of points by straight lines. We have studied statistical properties of the rescaled ISCT paths based on the numerical data of large scaled computer simulations.
iv.1 Fractal Dimensions
Figure 7 shows a log-log plot of the box counting of segments of with respect to the box sizes for with . As shown by this figure, the data for any can be fitted by a straight line very well and we can evaluate the approximate values of fractal dimensions for finite .
The evaluated values up to at most are then plotted versus in Fig.8. For each evaluation we prepared samples, and the ranges of scattering of results are shown by error bars in the figure. There limits are extrapolated by three-parameter fittings ; . The obtained values by the extrapolation are denoted by .
Figure 9 shows dependence of on . The Hausdorff dimensions of the SLE curves given by (4) are also shown by a dotted line. Systematic deviation is found between for the SLE curves and numerically evaluated for the rescaled ISCT paths. We observe in Fig.8 that the approximate value of fractal dimension for finite is increasing as is increasing, and the ratio of increment becomes larger as approaches the value 8. So we expect that the deviation will be systematically reduced, if we can perform numerical simulation for larger ’s and make appropriate extrapolation to the limit.
iv.2 Generalized Cardy’s Formula
Here we first consider the SLE curve in the case
Then, is the leftmost point in the interval , at which hits . Note that, if , , and if , , with probability one. For (32), has a nontrivial distribution. For each , we observe whether or . Figure 10 illustrates the former case. For the SLE curves with (32), the following formula is established. (See Proposition 6.34 in Law05 ().)
where is the gamma function and is Gauss’s hypergeometric function. This formula can be regarded as a generalization of Cardy’s formula, since the original formula corresponding to (34) with was derived by Cardy Car92 () for the “percolation exploration process” in the critical percolation model, and then the continuum limit of that process was proved to be described by the SLE curve with Smir01 (); Bef07 (). For , the above formula gives a power law
with the exponent
Now we consider the ISCT paths. For integers and with , prepare a realization of RW represented by . For each , by using the data , we calculate the position in following the recurrence formula (24). As noted at the end of Sec.III, for any . So we set a small value and look for the event
If , we define and calculate the value . If , that is, the event (37) does not occur for the given , then . The probability distribution function for the rescaled ISCT paths, which corresponds to , may be given by
In numerical calculations, we have set and and prepared realizations of RW, which are randomly generated. Since , we are allowed to consider only small values of and . The threshold value is wanted to be small, but for finiteness of , it should be positive. In Fig.11, we show log-log plots of the numerical evaluations of for with , with , and with . For finiteness of and smallness of the number of samples , data scatter for small . We find, however, power-law behaviors in the intermediate regions of ;
By linear fitting as shown in Fig.11, we have evaluated the values of exponent . The results are plotted in Fig.12, where Eq.(36) derived from the generalized Cardy’s formula is also shown by a curve. The good agreement implies that in the proper limit (39) will also follow the generalized Cardy’s formula in the parameter region (32).
V Concluding Remarks
In the present paper, we have proposed an algorithm, which generates a random discrete path on the upper half complex-plane as a functional of a path of random walk on the real axis for integers and . The system has one parameter and we call the path an ISCT path. We have studied the rescaled ISCT path defined by for large by computer simulations. The numerical analysis of the distributions of supports our expectation that the limits of the rescaled ISCT paths will have the same statistical properties as the SLE curves have.
The rescaled ISCT paths can be regarded as discrete approximations of the SLE curves. In this sense, the present study could be included by the previous numerical work Ken07 (); Ken09 (). In this paper, however, we have emphasized on our interest in the ISCT itself as a simple algorithm to generate complicated discrete dynamics of a point on . Dependence on the parameter of complexity of the SLE curves is demonstrated by dependence of complexity of the network of the ISCT paths on the angle of the slit generated by a single SCT.
We have learned that stochastic analysis is necessary and useful to study statistics and stochastics of the SLE curves Law05 (); Law07 (). Although we have reported only numerical study in this paper, we hope that the combinatorics and statistical mechanics methods developed for solvable models on lattices will be useful to analyze statistics and stochastics of the ISCT paths on , since they are functionals of simple random walks in one dimension.
Acknowledgements.The present authors would like to thank M. Matsushita and N. Kobayashi for useful discussion on application of the fractal analysis to the present work. This work was partially supported by the Grant-in-Aid for Scientific Research (C) (No.21540397) of Japan Society for the Promotion of Science.
Appendix A Recurrence Relations
Let for . Then (28) gives
When we set , the above gives
- (1) O. Schramm, Israel J. Math. 118, 221 (2000).
- (2) G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 2005).
- (3) G. Lawler, e-print arXiv: 0712.3256 [math.PR].
- (4) W. Kager and B. Nienhuis, J. Stat. Phys. 115, 1149 (2004).
- (5) J. Cardy : Ann. Phys. 318, 81 (2005).
- (6) M. Bauer and D. Bernard, Phys. Rep. 432, 115 (2006).
- (7) C. Pommerenke, Univalent Functions, (Vandenhoeck & Ruprecht, Götingen, 1975).
- (8) J. Cardy, e-print arXiv: math-ph/0103018.
- (9) M. Bauer and D. Bernard, Phys. Lett. B 557, 309 (2003).
- (10) R. Friedrich and W. Werner, Commun. Math. Phys. 243, 105 (2003).
- (11) Loewner considered conformal maps from a unit disk to one-slit domains with a slit starting from a point on the circle , and used a different equation from (2). In the present paper, we consider a modified version suitable for and discuss the chordal Loewner equation (LE) rather than the original radial LE.
- (12) Usually the time evolution of the inverse map