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Self-Avoiding Random Loops IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 6, NOVEMBER 1988 1509 Self-Avoiding Random Loops LESTER E. DUBINS, ALON ORLITSKY, JIM A. REEDS, AND L. A. SHEPP Abstract-A random loop, or polygon, is a simple random walk whose formly distributed random element of the set of self-avoid- trajectory is a simple closed Jordan curve. The study of random loops is ing n-walks. Equivalently, it is a sequence of independent extended in two ways. First, the probability P,(x, y) that a random n-step random variables So,. , S, ~ 1, where Sj = 1, each loop contains a point (x, y) in the interior of the loop is studied, and * + i, p,?(1/2,1/2) is shown to be (1/2)-(1/n). It is plausible that P,(x, y) /1 with probability 1/4, conditioned so that T, # Tk for all 1/2 for all (x, y), but we do not know this even for (x,y) = (3/2,1/2). 01 j<kIn, where To=O and ?=So+ +SjP1. Second, a way is offered to simulate random n-step self-avoiding loops. An n-walk s (or t) is a self-avoiding n-loop if tj # t, for Numerical evidence obtained with this simulation procedure suggests that all 0 I j<k < n and to= t,. Necessarily, n is even. A P,(3/2,1/2) = (1/2) - (c/n).(Here is where Steve Rice’s particular magic random self-avoiding n-loop is a uniformly distributed would have been especially useful.) random element of the set of self-avoiding n-loops. Equiv- alently, it is a sequence of independent random variables DEDICATION So,-.., Sn-l,where S, = +1, * i, each with probability 1/4, conditioned so that T, f T, for all 0 5 j < k < n, and T, = TEVE RICE was truly admirable. He was so gentle 0, where To = 0 and T, = So + - + SjPl. Sthat he disdained to use the word “no” in conversa- As the list of references of this paper suggests, many tion, especially if you were wrong. Among his many math- papers treat self-avoiding random walks and loops. Addi- ematical skills were a deep knowledge of special functions, tional references, undoubtedly hundreds of them, can be a prehistoric insight into Gaussianity, and a magical ability obtained by using “self-avoiding” in a key-word search of for numerical analysis. the Math Reviews or Science Abstracts databases. Many of these papers arise in the study of polymer chemistry and I. INTRODUCTION statistical mechanical models, e.g., Helfand and Pearson Walks on the two-dimensional rectangular grid corre- [l],Simon [2], and Chayes [3]. spond in an obvious way to walks on the Gaussian integers For papers of mathematical interest, see Hammersley in the complex plane: the complex numbers a + ib where [4],~Kesten pj, Guttmann- 161, LeFaii-vj, Tippenger a and b are integers and i is the square root of -1. For [8], and the references therein. Among other results, notational convenience, we describe the walks in the latter Hammersley [4] has shown that the number W, of self- setting. avoiding n-walks satisfies (l/n) log W, -, log p and that A step is an element of (1, -1, i, - i}, thought of as a the number L, of self-avoiding n-loops also satisfies unit step right, left, up, or down the complex plane. A (l/n)log L, + logp (amazingly, the same p). He proved walk is a sequence of steps. The length of a walk is the these facts by clever subaddivity arguments. Kesten [5] number of steps in the sequence. A walk of length n is an gave a functional-type equation for p and Guttmann [6] showed that = 2.6381 0.0002. n-walk. Let so,.. e, snP1 be an n-walk. Its trajectory is the p * sequence t,(s),. e, t,(s) defined by to(s)= 0 and, for Let to,. ., t, be a self-avoiding n-loop. The polygon 11 jsn, tj(s)=s,+ ... +sJP1. It is the sequence of formed by the points to,. e, r, along with the unit seg- points in the complex plane visited by a walker who starts ments connecting t, to tj+l for j = 0; . -,n - 1 form a at zero and follows the steps so,-. ., s,-~. For brevity, we simple closed, or Jordan, curve in the plane. Such a curve write tj for t,(s) and use s and t interchangeably to divides the plane into an inside region and an outside describe the same walk. region. For any point (x, y) in the plane, we define An n-walk s (or t) is selfauoiding if t, # t, for all P,(x, y) = P((x, y) E inside the region 01 j<k 5 n. A random self-avoiding n-walk is a uni- of a random n-loop) In Section 11, we prove that Manuscript received January 1988; revised February 24, 1988. This work was supported in part by the National Science Foundation under Grant DMS 86-01634. This paper was presented at the Midwest Regional Meeting of the Institute of Mathematical Statistics, Madison, WI, May 16-18, 1988. hence P,(1/2, 1/2) 71/2 as n + CO. We conjecture that, L. Dubins is with the Department of Mathematics, University of whenever x and y are both nonintegers, P,(x, y) P 1/2 as California, Berkeley, CA. n + CO. However, we cannot prove it even for (x, y) = A. Orlitsky, J. Reeds, and L. A. Shepp are with AT&T Bell Laborato- ries, 600 Mountain Avenue, Murray Hill, NJ 07974. (3/2,1/2). Even proving that P,(3/2, 1/2) I P,(1/2, 1/2) IEEE Log Number 8824880. seems difficult, so in Section IV we resort to simulations. 0018-9448/88/ll00-1509$01.00 01988 IEEE Authorized licensed use limited to: University of Pennsylvania. Downloaded on March 17,2010 at 13:33:21 EDT from IEEE Xplore. Restrictions apply. 1510 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 6, NOVEMBER 1988 The straightforward simulation technique for random Consider the n Jordan curves J,, fl; * -,J,-l obtained by self-avoiding n-loops that simply walk to a random near- cyclically permutting the vertices to - T,, T,,, - est-neighbor not previously visited, then returns to the ?,...,T,+,-T/ forj=O,l;..,n-l, where T,+i=T for origin encounters two problems. First, one may reach a 0 Ii < n. Consider also the 4n Jordan curves obtained by position that has no such nearest neighbors. According to rotating each of these n Jordan curves by 0", 90", 180", [9] and [lo], this happens after an average of about 71 270" around the point 0. Some of these may be identical, steps. Thus, especially for large n, this simulation and but J may be any of these with equal probability. Once it others (for instance, [11]-[15]) would not seem very useful. is shown that the fraction f of these which contain A second problem is ensuring that the walk returns to the (1/2,1/2) as an inside point is (1/2) - (l/n), it will follow origin after exactly n steps. that Instead, this paper proposes a loop-generation algorithm 11 that performs a random walk on a graph of all self-avoid- P( ( f , f ) is inside J) = f = - ;. (2) ing n-loops with a certain edge structure. The resulting process is a Markov chain, whose states are the self-avoid- To prove that, suppose that vertex T, of J is convex. Then ing n-loops. The edge structure of the graph and transition exactly one of the four rotations of J/ contains (1/2,1/2). probabilities are chosen so that the equilibrium distribu- If T/ of J is concave then exactly three of the four tion of the Markov chain, hence of the derived loop, is rotations of J, contain (1/2,1/2), and if SJ is straight, uniform. then exactly two of the four rotations of JI contain This technique is similar to, and draws heavily from, the (1/2,1/2). Therefore, pivot algorithm used to generate random walks (rather u+3c+2s than loops). The pivot algorithm, invented by La1 [16], f= 4n . performs a random walk on a graph of all self-avoiding n-walks. It has been used by Olaj and Pelinka [17], Since the total signed curvature of a simple closed planar MacDonald et al. [HI, Hunter et al. [19], Stellman and polygon is f2a it follows that Gans [20], [21], and Freire and Horta [22]. A neat analysis u-c=4 of its performance is given in a recent paper by Madras and Sokal [23]. For completeness and clarity, their descrip- and so, by (1) and (2). tion of the pivot algorithm is duplicated in Section 111, and 2u+2c+2s+c-4 1 1 =--- the loop algorithm is given in Section IV. f= 4n 2 n' The number of iterations required to reach equilibrium in both the loop and the pivot algorithm seems to be small The above proof shows that the fraction of rotations and (perhaps linear or quadratic in n). However, there is no cyclic permutations of any n-loop which contain (1/2,1/2) known proof that a polynomial number of iterations suf- is (1/2)-(1/n). It is worth noting that one can prove fices for either the pivot or the loop algorithm. somewhat more: that the rotations can be ignored. The translation-shape of an n-loop J with nodes T,;..,T, is the set S of n n-loops J-T,, l_<j<n, 11.
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