Sampling Self Avoiding Walks

Transcription

1 Samplng Self Avodng Walks James Farbanks and Langhao Chen December 3, 204 Abstract These notes present the self testng algorthm for samplng self avodng walks by Randall and Snclar[3] [4]. They are ntended to be used towards the end of a graduate class on Markov Chans. We are solvng the problem of samplng Self Avodng Walks of a fxed length. The man lessons to learn are.. In Barett-Sokal chan, changng a fxed constant to a sequence of constants allows one to prove convergence. 2. The mxng rate of the algorthm s polynomal f c jc k apple c j+k g(j + k) for some polynomal g. 3. We can fnd the necessary,a by bootstrappng. 4. For any g we can test the necessary conjecture. Introducton A walk s self avodng f no vertex s vsted more than once. We want to sample self avodng walks n order to answer physcal questons about polymers such as how many possble walk are there of a gven length, and how far s the free endpont from the orgn n a typcal SAW. Let c n be the number of self-avodng walks of sze n. We can count lattce walks exactly. At each step there are 2d possble neghbors and at each step every neghbor s a vald transton so there are (2d) n possble walks of length n on the d dmensonal lattce. Snce there are no teractons between choces, we can sample exactly from the unform dstrbuton on lattce walks wthout a markov chan. We cannot sample random walks wth the process that starts wth a walk of length and extends t wth equal probablty n any drecton that does not create a self ntersecton, because at each step some choces of drecton wll lead to more possbltes at longer steps. The pvot algorthm analysed n [2] ntroduces a Markov Chan that uses very nonlocal moves that have not too small acceptance fracton. The authors make a heurstc argument about the autocorrelaton tme, but do not prove fast mxng. The pvot algorthm takes steps where each vertex of the walk s chosen at random and then a randomly rotataton or reflecton s appled to the segment after ths pont. These steps take ponts very far n dstance but are not proven to have fast mxng. It s clear that cn+m c nc m apple because t s the probablty of pckng one walks of length n, m and concatenatng them to form a self avodng walk of length n + m. 2 The Markov Chan Construct Markov Chans M,M 2,..., where each M n has state space n = [ n =0 S the unon of all self avodng walks of length at most n. The physcs communty beleves that there should be one parameter that controls the MC but ther chan cannot be proven to mx rapdly. Relaxng ths to a sequence of parameters s allows us to prove rapd mxng and unformty for the correct choce of.

2 For a self-avodng walk w 2 n of length apple n, choose one edge randomly from the 2d edges that are ncdent to the free endpont of w. If ths edge concdes wth the last step of w, remove ths edge. If addng ths edge also makes a self-avodng walk of length at most n, add ths edge wth probablty +. Defne w w 0 < and the frst steps of w 0 concdes wth w. Defne w w 0 w w 0 and = +. Statonary dstrbuton s 8 w >< 0 /2d f w w 0 P n (w, w 0 /2d f w 0 w )= r(w) f w = w >: 0 0 otherwse n (w) = Z n = for w 2 n where Z n s the normalzng factor P Q w2 n = s chosen as c /c so that the statonary dstrbuton s unform on the levels and unform wthn the levels. Physcsts beleve that there s a lmt for c /c so Barett and Sokal chose a sngle for all c /c. Here s replaced by a sequence of s whch can be determned by bootstrappng. Ths allows the algorthm to provde correct answers at each level even f there were no constant that satsfed the conjecture at all of the levels smultaneously. 2. Statonary Dstrbuton Here are the detals of the statonary dstrbuton that were skpped n class. f w< w 0 : f w 0 < w: f w 0 = w: = = = 2d = 2d = r(w) = = = = 2d 2d r(w) otherwse both transton probabltes are 0 so the detaled balance equaton s satsfed. Assume that each s equal to c /c.then 2.2 Unform n (w) = c = Z n c = Z n c Here are the detals of the unformty that were skpped n class. Snce the statonary probablty s the same for each walk of the same length, condtonng on the length gves the unform dstrbuton on that length. The dstrbuton of lengths s gven by the followng where ˆj s the set of paths wth length exactly j w 0 extends w P ( = j) = Z n ˆj j = 2

3 Ths tells us that f each of the products Q j = s equal to the number of walks wth length j, thenthe dstrbuton s unform on all walks. Ths can be acheved by settng = c c because the product telescopes approprately. We mght be nterested n basng towards large walks, whch would lead to values that are larger. We also cannot a pror state the number of walks of a gven length snce ths s the quantty we are tryng to estmate. If were known, then we could run the markov chan and sample e cently. Snce we need to fnd the, we bootstrap ths parameter. The nnovaton of allowng the parameter to vary at each level of the tree allows the sampler to work for all levels up to the pont where the conjecture s false, f such a pont exsts. The Barett and Sokal chan wth a fxed for all levels s not robust n ths way. 3 The Mxng Tme (exact case) We defne a sequence n whch bounds the fracton of pars of self avodng walks whose concatenaton has length n whch are self avodng. c j+k n = mn j+k<n c j c k The key dea s to bound the congeston across any edge at length k. WehaveQ(e) = n (w)p n (w, w 0 )= /4dZ n c k+ for the ergodc flow across any edge. and the probablty of any subtree S whch s the extensons of w 0 wth length k +s n (S) = X w w 0 n ( w) () = apple apple Z n c k+ Z n c k+ nx j=k+ nx j=k+ c k+ c j extensons of w wth j steps (2) c k+ c j k c j (3) n Z n c k+ n (4) Where we use the fact that the number of ways to extend a walk by steps s less than c whch s the sub-cayley property for the tree. So ( ) apple 4dn n. From the mxng tme bound based on congeston we get a factor of log n (0) = log n and a factor of 2n from the length of the longest path n a tree. So the fnal mxng tme s O dn 2 n log n 4 Assumed Conjecture Physcsts beleve that there s a lmtng constant for the expanson probablty c /c, and that c n = µ n poly(n). It s wdely beleved that for any dmenson d, there exsts fxed polynomal g such that, for all n, m, c n c m apple g(n + m)c n+m (5) That s that the probablty of successful concatenaton s bounded below by /g(n). The conjecture mples that n apple g(n). The bound on the mxng tme of the Markov Chan wll be presented n terms of. So ths conjecture wll mply fast mxng. 5 Bootstrappng The mxng tme of M n depends on n and the transton probabltes depend on. We know nether of these. In fact the are the prncple quantty of nterest to understandng the set of random walks. So we wll assume that we can sample from M n and show how to estmate n and n. 3

4 To estmate the s we generate t n ndependent pars of walks from S S n for each apple n. Then we count what fracton of pars of them produce self avodng concatenatons, call ths q n,. By takng the mnmum of n and q n, we get an estmator wthn a factor of 4 of the true answer wth su cent probablty. The 0/ estmator theorem says that t n = O an log n/ s su cently large for desred falure probablty. Ths leads to a Õ(n4 n 2 ) cumulatve runnng tme. One factor of n for the per sample tme and another factor from the number of samples. We can start wth the known quanttes =/2d, c =2d =. To estmate the n = c n /c n,we generate samples from M n and then count what fracton can be extended to length n Then our estmate for the number of walks at length n s c n / n and we use ths value of n to estmate the n. 6 Self Testng Everythng sad about the chan s true regardless of the conjecture gven. The bounds are n terms of n whch mght grow superpolynomally. If we want to guarantee that the runnng tme s polynomal then we need to know that n s polynomally bounded. Snce we have not made progress on the conjecture, we adapt the algorthm to self-test. Gven any functon g, we can test the conjecture c m c n apple g(m + n)c m+n. We run the estmate process above and then check f > 4g(n). If t s, then the conjecture has a counterexample wth hgh probablty. If our estmates always pass the check, then we can be confdent that the answers are relable wth hgh probably. The algorthm can output ether a relable answer or a error message wth probablty. 7 Later Work n The followng are three possble drectons that later work can be bult based on ths paper.. Conjecture: For any dmenson d, there exsts fxed polynomal g, 8n, m c n c m apple g(n + m)c n+m Provng ths conjecture wll make ths algorthm the frst polynomal tme Monte Carlo approxmaton algorthm for self-avodng walks. 2. There can be other algorthms that are self-testng lke ths one. When an algorthm s based on a conjecture, lettng the algorthm reject f the conjecture s false wll make t a more robust algorthm. 3. There are a few areas n physcs and bology where the result n ths paper can be appled to. Gambn and Wojtowcz[] appled ths algorthm on lattce models of proten foldng. Proten foldng s a physcal process where a proten changes ts three-dmensonal structure from random col. Whle there are several lattce models for proten, Gambn and Wojtowcz used the smplest one called three-dmensonal HP model, where each proten s represented by a sequence of H s and P s ( H represents hydrophobc amno acd type and P represents polar amno acd type). They consdered these sequences as self-avodng walks. Instead of defnng c n as the number of self-avodng walks of length n, they defne t as the sum of weghted self-avodng walks of length and ntroducng weghtng factor. c n = P w2s h(w) They made up two HP sequences and expermented by the EstmateBeta method. They got a table of numercal results and observed a pattern: The quantty n seems ndependent wth and the hydrophobcty pattern of the sequence. 4

5 References [] Anna Gambn and Daman Wójtowcz. Almost fpras for lattce models of proten foldng. In Expermental and E cent Algorthms, pages Sprnger, [2] Neal Madras and Alan D Sokal. The pvot algorthm: a hghly e cent monte carlo method for the self-avodng walk. Journal of Statstcal Physcs, 50(-2):09 86, 988. [3] Dana Randall and Alstar Snclar. Testable algorthms for self-avodng walks. In Proceedngs of the Ffth Annual ACM-SIAM Symposum on Dscrete Algorthms, SODA 94, pages , Phladelpha, PA, USA, 994. Socety for Industral and Appled Mathematcs. [4] Dana Randall and Alstar Snclar. Self-testng algorthms for self-avodng walks. Journal of Mathematcal Physcs, 4(3): ,