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How a random walk covers a finite lattice
Physica A 185 (1992) 35-44 North-Holland
H o w a r a n d o m w a l k c o v e r s a finite lattice M.J.A.M. B r u m m e l h u i s a a n d H . J . H i l h o r s t b ~Instituut-Lorentz, Rijksuniversiteit te Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands bLaboratoire de Physique Th~orique et Hautes Energies 1, Universit~ de Paris-Sud, bhtiment 211, 91405 Orsay, France A random walker is confined to a finite periodic d-dimensional lattice of N initially white sites, When visited by the walk a site is colored black. After t steps of the walk, for t scaled appropriately with N, we determine the structure of the set of white sites. The variance of their number has a line of critical points in the td plane, which separates a mean-field region from a region with enhanced fluctuations. At d = 2 the critical point becomes a critical interval. Moreover, for d = 2 the set of white sites is fractal with a fractal dimensionality whose t-dependence we determine.
1. Introduction T h e t r a d i t i o n a l c o n n e c t i o n b e t w e e n r a n d o m walks a n d fractals is that a r a n d o m walk o n a lattice g e n e r a t e s a fractal of fractal d i m e n s i o n 2. T h e s t a t e m e n t holds for a r a n d o m walk whose l e n g t h b e c o m e s infinite, a n d the fractal r e f e r r e d to is the set of all lattice sites visited by the walk; o n e m a y i m a g i n e these to be c o l o r e d b l a c k in a white b a c k g r o u n d . T h e fractal so d e f i n e d is n o n t r i v i a l o n l y in spatial d i m e n s i o n s d larger than 2: for d = 2 a n infinite r a n d o m walk will b l a c k e n each lattice site with p r o b a b i l i t y o n e . Here
we shall e x h i b i t - a m o n g a few o t h e r t h i n g s - a different r e l a t i o n
b e t w e e n a simple r a n d o m walk a n d fractals. W e c o n s i d e r a r a n d o m w a l k e r , c o n f i n e d to a finite p e r i o d i c lattice of N sites, which b l a c k e n s all sites that it visits. A n e x a m p l e is s h o w n in fig. 1. W e t h e n ask what the p r o p e r t i e s are, after t steps, of the set of white sites. I n p a r t i c u l a r , is t h e r e a n y fractal b e h a v i o r ? T h e set of sites in which we are i n t e r e s t e d is c o m p l e m e n t a r y to the t r a d i t i o n a l o n e , a n d t h e a n s w e r a p p e a r s to be, too: fractal b e h a v i o r occurs only in d i m e n s i o n 2. W e shall see, n e v e r t h e l e s s , that in h i g h e r d i m e n s i o n s t h e r e are also i n t e r e s t i n g phenomena.
Laboratoire associ~ au Centre National de la Recherche Scientifique. 0378-4371/92/$05.00 (~) 1992- Elsevier Science Publishers B.V. All rights reserved
36
M.J.A.M.
Brummelhuis, H . J . Hilhorst / H o w a random walk
I
covers a finite lattice
I
• milS11
mm~ IIIIIID IIIIIL:
mmw
Ill[
~is~,~
~!
Fig. 1. A square lattice of N = 40 × 40 sites. A random walker has started in the origin and performed t = 4880 steps. All sites visited are colored black. The theory of this paper, eq. (22), gives for the fractal dimension of the set of white sites (on a scale small compared to the linear lattice size) the value d v = 1.82.
2. T h e lattice c o v e r i n g t i m e and the last-site probability Since e v e n t u a l l y w e shall n e e d to consider the limit o f infinite lattice size, N - - ~ ~ , w e m u s t specify the relation b e t w e e n t and N. Interesting behavior, like fractality, has a c h a n c e of occurring o n l y if the "time" t scales in such a w a y that the total number of white sites W(t) is m u c h less than N. H e n c e a p r e l i m i n a r y q u e s t i o n to be a n s w e r e d is h o w t has to d e p e n d on N in order that this be the case. W e shall do so by first studying the f o l l o w i n g t w o questions.
Question 1. W h a t is the t i m e ~'u at which the last site is visited? W e shall call 'rN the lattice covering time; it is a stochastic variable described by s o m e (in g e n e r a l u n k n o w n ) probability distribution. Question 2. W h a t is the probability LN(x ) that the last site is x (and not s o m e o t h e r site x ' ) ? W e shall refer t o LN(X ) as the last-site probability. 2.1. Mean field and dimension d = 1 It is instructive first to study a m e a n field or " d = ~ " m o d e l [1]. A w a l k e r starts at the origin, x = 0 , of a lattice of N sites, and m a k e s a j u m p to a r a n d o m l y s e l e c t e d site out of the N - 1 u n o c c u p i e d o n e s each unit of time. Let
M . J . A . M . Brummelhuis, H.J. Hilhorst / H o w a random walk covers a finite lattice
37
P(W, t) be the probability that after t jumps there are still W whites sites left. It is easy to write down and solve a recursion in time for this quantity; the result is W(t)
=--~ WP(W,
t) = N e -'IN
as N----> oo.
(1)
W
T h e last site is visited when W(t)= ~7(1), which happens when t = N log[N/ G ( 1 ) ] - N log N. Upon identifying this time with the average lattice covering time ~N we see that
~N=NlogN
for
d=~,
N---~.
(2)
This identification rests upon the replacement of the average of the time evolution by the time evolution of the average. In the mean field case it can be justified by an independent calculation of ~N; in general, it should be correct when P(W, t) is sharply peaked in W, a property that we shall investigate below. Since all sites in the mean field model are equivalent, we have trivially that 1
LN(X ) =
N-1
for x # 0
(3)
(mean field). 0
for x = O
The mathematician Aldous [2] showed in 1983 that on hypercubic lattices with periodic boundary conditions
7rN=AdNIogN
for
d=3,4,5 ....
and
N---~o0.
(4)
This work provides expressions for A d in terms of the probability of return to the origin, and finally leads to [1] A 4 =
1.239
....
A 3 = 1.516 . . . .
(5)
These values were confirmed in computer simulations by Nemirovsky, Mfirtin and Coutinho-Filho [3], who fit their data on two-dimensional lattices to
~s = A2Nlog 2N ,
d = 2,
N----~~ ,
(6)
wit an estimated A 2 = 0.33. It is amusing also to consider the one-dimensional case, where the periodic
38
M.J.A.M.
B r u m m e l h u i s , H . J . Hilhorst / H o w a random walk covers a finite lattice
lattice is a closed loop of N sites. It was shown by Yokoi, Hernfindez-Machado and Ramirez-Piscina [4] that "~U= ½N(N- 1) for d = 1. (Interesting results for the average time needed to visit all sites of a loop at least k times come from simulations by Mirasso and Mfirtin [5].) Subsequently Brummelhuis and Hilhorst found that the last-site probability distribution LN(x ) in d = 1 is given by exactly the same expression, eq. (3), as in the mean field case, a result which at first sight may be somewhat counterintuitive!
2.2. Dimension d >!2 One might now guess that LN(x ) is given by eq. (3) in all dimensions 1 ~< d <~ 2. It is not obvious, however, how to determine LN(X ) for arbitrary dimension by the standard methods of random walk theory. The reason is that " x being the last site visited" is a property that refers to the walker's whereabouts during its whole history. There is, in particular, no simple differential equation from which LN(x ) may be solved. One can of course write LN(x ) in terms of a path integral on all walks, but the resulting expression remains formal for 1 < d < 2. We shall now show how LN(x ) can be obtained, for all d ~> 2, by a method that is based on a separation of time scales. In the course of our argument we shall also prove that the two-dimensional lattice covering time is indeed given by eq. (6), and that A 2 = ,rr
1=0.318 . . . . .
(7)
Let Wt(x ) be the variable that equals 1 if at time t site x is still white, and 0 otherwise. Its average W,(x) is the probability that at time t site x is still white. Let furthermore F,(x) be the first passage time distribution for site x, i.e. the probability that the first visit of site x will take place exactly at time ~-. Then
(8)
W,(x) = 1 - ~ F~(x). ":,"= 0
Standard methods of random walk theory express F,(x) in terms of the random walk G r e e n function. We refer to ref. [1] for the relevant formulas for a hypercubic lattice with periodic boundary conditions. Upon substituting these known results in eq. (8) one arrives at
1
Ca 2 ) e -'/AdN ]x-~-_
for d > 2
(9a)
for d = 2 ,
(9b)
W,(x) =
2 ~log Ixl e - ,,N,ogN
39
M.J.A.M. Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
where C d is a known constant. Both eqs. (9) are valid for N---> o0 and Ixl l, with t at least of order N in (9a) and at least of order N log N in (9b). For x near the periodic boundary of the lattice there are corrections [1]. We first note that by summing eq. (9) on all x one obtains the average total n u m b e r of white sites still left at time t, that is, the quantity W ( t ) discussed before. If one looks for the average lattice covering time ~N by setting this n u m b e r equal to t?(1) and solving to leading order in N, one finds that in dimensions above two, eq. (9a) leads to (4) and that for dimension two eq. (9b) leads to (6) together with (7). Hence for d = 2 we have confirmed the conjecture (6) by Nemirovsky et al. [3] and calculated the constant A 2. This completes the answer to question 1. More detailed information comes from the prefactors in eq. (9). These show that at any given time different sites have different probabilities of still being white, but that these probabilities have time-independent ratios for t of order N or larger, when d > 2, and for t of order N log N or larger, when d = 2. We claim now that the prefactors in (9), when multiplied by the overall normalization factor N - i , are equal to the last-site probabilities LN(X ). The justification is as follows. For large times (t increasing faster than N 2/d) the probability distribution of the random walker position is uniform over the lattice. Therefore all sites which are still white at such times have the same rate of decay to the black state. This explains the x - i n d e p e n d e n c e of the exponentials in eq. (9). That the prefactors in (9) do depend on x is therefore due to a transient effect, namely the fact that the walker starts at t = 0 at the precise site x = 0. H e n c e , in addition to the uniform decay discussed above, sites near the origin have an extra probability of being colored black by the random walker on its initial path away from the origin. Indeed one observes that L N ( x ) is depressed near the origin, and approaches the value N -1 for large x from below. This completes the answer to question 2. 3. Fluctuations on time scale 7N
We now introduce a new time variable a and consider times t such that t = o~N .
(10)
U p o n substituting (10) in (9), using (4) and (6), and summing on x (which just amounts to replacing the prefactor by N), one finds that the total number of white sites still left for t = a~ N and N large is W(a~N) = N 1-~ .
(11)
40
M.J.A.M.
B r u m m e l h u i s , H . J . Hilhorst / H o w a random walk covers a finite lattice
For 0 < a < 1 and N--* ~ this number is much less than the initial number, N, but still much larger than 1. For l < a one should interpret eq. (11) as indicating that with a large probability the lattice is fully black, but that with a small probability some white sites are left. We shall want to know first how sharply the distribution P(W, t) is peaked. We therefore ask the following question.
Question 3. What are the f l u c t u a t i o n s AW2(ti for times t OliN? We remark that if each site were colored white independently with probability N -s, one would have A W 2 1 / 2 = N ~1-~)/2= lye 1/2. Secondly, we would like to know something about the clustering properties of the white sites. We therefore ask the following question, which is a test for fractality. =
Question 4. What is the number Wt(RIx ) of white sites within a distance R from a given white site x? Questions 3 and 4 both involve the correlation between pairs of sites. The key role in answering them is played by the correlation function
AW,(x, y) =- W,(x) Wt(y ) - W,(x) W , ( y ) .
(12)
By a relation which generalizes eq. (8), one can express Wt(x)W,(y) in terms of the probability F~(x, y) that the first visit of the random walker to the set of two sites {x, y} takes place exactly at time r. Since F~(x, y) can again be expressed, by standard methods, in terms of the Green function of the random walk, one obtains an expression for AW,(x, y) on the same footing as eq. (9).
3.1. Dimension d > 2 The properties of AW,(x, y) relevant for our purpose are, for N---~ ~ and a fixed [1],
AW,(x, y) = W~(x) ,
d >12, (13a)
AWt(x, y) = N-2~[cl log N/Ix - yl a-2 + c 2 log 2 N/lx - y12d-4],
d >2. (13b)
(In (13b) we suppose x and y on the scale N l/d of the linear lattice size. Terms with x and y at a fixed distance # 0 are proportional to powers of N less than - a and need no special consideration. See ref. [1].) By summing AW,(x, y) on x and y one obtains the mean square fluctuations
M . J . A . M . Brummelhuis, H.J. Hilhorst / H o w a random walk covers a finite lattice
41
AW2(t). In this summation the terms with x = y and those with I x - Yl large compete for the dominant role. The leading order term in (13b) vanishes exactly [1] when one sums it on x - y and one takes into account the finite size and s m a l l - I x - y[ effects; it is therefore not of interest here. Fig. 2 helps to represent the results for the mean square fluctuations. One finds, also using eq. (9a) for W,(x), that the a d - p l a n e is divided into two regions by the line a = 4/d - 1. Explicitly, AW-(OtTN)I/2=N (1- a ) / 2 AW2(oITFN)I/2
oc N 2 / d - a
~,/1/2
=
for
a>4/d-1,
d>2,
(14a)
log N = (1 - a) -1 1~~2/d-")/cl-~) log 15' for
a<4/d-1,
d>2.
(14b)
T h e fluctuations of eq. (14a) are as for independently colored sites. The calculation actually shows that it stems solely from the x = y terms (eq. (13a)) in the summation, the x ~ y terms of eq. (13b) constituting a negligible correction. Above the critical dimension d = 4 the correlations (13b) are not integrable at the origin and the result (14a) holds [1] for all times a > 0 , one may show that it also holds for the d -- ~ model discussed above. In eq. (14b), since d > 2 , the power of W is larger than ½ but less than 1. This strong enhancement of the fluctuations AW2(t) is due to the dominance of correlations (eq. (13b)) between the most distant pairs of sites x and y. The interpretation is as follows. Initially, that is, for not too large a, the white regions in the lattice are large and the fluctuations AW2(t) are dominated by the fluctuations of their sizes. As time goes on these large regions are broken
'I
d
4~
MF
3
I ,
,
0 Fig. 2. Representation of the properties of the set of white sites in the time(a)-dimension(d) plane. In the region marked LR the fluctuations in the total number of white sites are anomalously large due to long-range pair correlations; in the one marked MF the white sites are essentially randomly distributed, as in mean field. At d = 2 the line of critical points expands into a critical interval along which the exponent fl varies from ½ to 0.
42
M.J.A.M. Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
up into smaller ones, and beyond a critical time a¢ = 4 / d - 1, the fluctuations AW2(t) are dominated by the fluctuations in the n u m b e r of individual white sites. C o m m o n to both regions of eq. (14) is that for a < 1 one has AWa(t)l/2/ W(t)---~O as N - - ~ with t = a~ N, so that the distribution P ( W , t) is sharply p e a k e d for all d > 2. This answers question 3 for d > 2. 3.2. D i m e n s i o n d = 2
F r o m eq. (13b) it is clear that the lower critical dimension d = 2 must be special. Instead of eq. (13b) we now have [1], in addition to eq. (13a), which remains valid,
I cU-Olx - Yl o
AW,(x,y)=[N_Zf( )
for [ x - y [ ~ 0 and fixed as N---~o~,
for
N-'/Zlx-y I fixed
as N---~oo.
(15a)
(15b)
T h e constant c and the function f are known; in particular [1], f ( ¢ ) oc ~ -4~
for ~:~0
(16)
and f ( ~ ) remains finite for ~ of order 1. One sees that for a < ½ expression (15b) can be s u m m e d on all x and y, and that all distances Ix - y[ on the scale contribute. The result is AW2(oFrN)
1/2 ~-
C(ot) N 1-'~ = C(ol) W ( t ) ,
0 < a < ½,
d = 2.
(17)
It shows that the width of P(W, t) is of the same order in N as the average W ( t ) itself. C ( a ) is a known constant which is easily shown to diverge as 1/( ½ - a)l/2 as a l '1 . For ½ < a but still a < 2 , one sees from (16) that eq. (15b) is no longer integrable at the origin, but eq. (15a) still is. H e n c e the behavior of A W , ( x , y) for 1 ~ Ix -- y[ ~ N 1/2 has to be analyzed m o r e precisely. This is done in ref. [1] and we state the result. T h e r e appears a time regime ½ < a < 2, represented by the heavy solid line in fig. 2, where the main contribution to W2(t) comes from pairs of sites with Ix - y [ on the scale N m~), in which ~(a) = 1 -V~7-2.
(18)
As a increases from ½ to 2, this spatial scale shrinks continuously from the linear lattice size N ~/2 to the lattice distance 1. The explicit result for the rms
M.J.A.M. Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
43
fluctuations in this time regime is A w e ( o z r u ) '/2 oc (log N) 1/4 N 3/2-V-~ ,
½
d=2,
(19)
and hence these are much larger than its average 1~' itself. For a = 2 the p o w e r of N in eq. (19) coincides with the one of the mean-field result (14a). This result continues to hold for ot > 2 (note however the interpretation of a > 1 given above), and is solely due to the x = y terms (eq. (13a)) in the summation. We have hereby completed the answer to question 3. We conclude that for d = 2 the distribution P ( W , t) is broad and nontrivial, and that different intervals of a must be distinguished. It remains unexplained that in spite of this broadness we should have been able to determine fN in good agreement with the c o m p u t e r simulations [3] by the argument following eq. (9).
4. Fractality Finally we address question 4 about the occurrence of fractality. The average n u m b e r of white sites within a distance R from a site x known to be white is
W,(Rlx) =
W t - ~ -1 ~ W t ( x ) W t ( x -~- r) r
= N - " K a R a + N " ~ AW~(x, x + r ) , r
(20)
where in the second step we have used (12) and chosen x on scale N ~/a, so that the prefactors from the initial effect can be ignored, and where K a is the v o l u m e of the d-dimensional unit sphere. With the aid of eq. (13b) it is easily checked that for all dimensions d > 2 the first t e r m in (20) dominates the second one as R ~ ~. H e n c e for d > 2 the white sites are statistically u n i f o r m l y distributed at large distances. For d = 2 the situation is different. For times 0 < a < 2 and R independent of N we can use (15b) in (20) and find that
W,(Rlx) ~ R 2-~
for R l a r g e ,
d = 2.
(21)
H e n c e in dimension d = 2 the white sites constitute a f r a c t a l of dimensionality dF=2D
o~
= 2 - ~ r t / N log 2 N
for t, N----~~ and a fixed.
(22)
44
M . J . A . M . Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
In ref. [1] it is shown that for ½ < et < 2 this fractal is essentially confined to a region of linear size N ~ ) , with f l ( a ) given by (18). T h e s e results p r o v i d e an a n s w e r to question 4. W e c o n c l u d e by e m p h a s i z i n g that for d = 2 there is a basic question which has b e e n c i r c u m v e n t e d and left u n a n s w e r e d in this work. We state it explicitly. Q u e s t i o n 5. W h a t is, in d i m e n s i o n d = 2, the distribution P(W, t) of the total n u m b e r W of white sites at time t? T h e interesting time regime is the one of eqs. (10), (7), and (6).
A n a n s w e r to this question would, a m o n g o t h e r things, explain o u r a p p a r e n t success in finding the m e a n lattice covering time ~ by an a p p r o x i m a t e argument.
References [1] [2] [3] [4] [5]
M.J.A.M. Brummelhuis and H.J. Hilhorst, Physica A 176 (1991) 387. D.J. Aldous, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 62 (1983) 361. A.M. Nemirovsky, H.O. Mfirtin and M.D. Coutinho-Filho, Phys. Rev. A 41 (1990) 761. C.S.O. Yokoi, A. Hernfindez-Machado and L. Ramirez-Piscina, Phys. Lett. A 145 (1990) 82. C.R. Mirasso and H.O. M~irtin, Z. Phys. B 82 (1991) 433.
H o w a r a n d o m w a l k c o v e r s a finite lattice M.J.A.M. B r u m m e l h u i s a a n d H . J . H i l h o r s t b ~Instituut-Lorentz, Rijksuniversiteit te Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands bLaboratoire de Physique Th~orique et Hautes Energies 1, Universit~ de Paris-Sud, bhtiment 211, 91405 Orsay, France A random walker is confined to a finite periodic d-dimensional lattice of N initially white sites, When visited by the walk a site is colored black. After t steps of the walk, for t scaled appropriately with N, we determine the structure of the set of white sites. The variance of their number has a line of critical points in the td plane, which separates a mean-field region from a region with enhanced fluctuations. At d = 2 the critical point becomes a critical interval. Moreover, for d = 2 the set of white sites is fractal with a fractal dimensionality whose t-dependence we determine.
1. Introduction T h e t r a d i t i o n a l c o n n e c t i o n b e t w e e n r a n d o m walks a n d fractals is that a r a n d o m walk o n a lattice g e n e r a t e s a fractal of fractal d i m e n s i o n 2. T h e s t a t e m e n t holds for a r a n d o m walk whose l e n g t h b e c o m e s infinite, a n d the fractal r e f e r r e d to is the set of all lattice sites visited by the walk; o n e m a y i m a g i n e these to be c o l o r e d b l a c k in a white b a c k g r o u n d . T h e fractal so d e f i n e d is n o n t r i v i a l o n l y in spatial d i m e n s i o n s d larger than 2: for d = 2 a n infinite r a n d o m walk will b l a c k e n each lattice site with p r o b a b i l i t y o n e . Here
we shall e x h i b i t - a m o n g a few o t h e r t h i n g s - a different r e l a t i o n
b e t w e e n a simple r a n d o m walk a n d fractals. W e c o n s i d e r a r a n d o m w a l k e r , c o n f i n e d to a finite p e r i o d i c lattice of N sites, which b l a c k e n s all sites that it visits. A n e x a m p l e is s h o w n in fig. 1. W e t h e n ask what the p r o p e r t i e s are, after t steps, of the set of white sites. I n p a r t i c u l a r , is t h e r e a n y fractal b e h a v i o r ? T h e set of sites in which we are i n t e r e s t e d is c o m p l e m e n t a r y to the t r a d i t i o n a l o n e , a n d t h e a n s w e r a p p e a r s to be, too: fractal b e h a v i o r occurs only in d i m e n s i o n 2. W e shall see, n e v e r t h e l e s s , that in h i g h e r d i m e n s i o n s t h e r e are also i n t e r e s t i n g phenomena.
Laboratoire associ~ au Centre National de la Recherche Scientifique. 0378-4371/92/$05.00 (~) 1992- Elsevier Science Publishers B.V. All rights reserved
36
M.J.A.M.
Brummelhuis, H . J . Hilhorst / H o w a random walk
I
covers a finite lattice
I
• milS11
mm~ IIIIIID IIIIIL:
mmw
Ill[
~is~,~
~!
Fig. 1. A square lattice of N = 40 × 40 sites. A random walker has started in the origin and performed t = 4880 steps. All sites visited are colored black. The theory of this paper, eq. (22), gives for the fractal dimension of the set of white sites (on a scale small compared to the linear lattice size) the value d v = 1.82.
2. T h e lattice c o v e r i n g t i m e and the last-site probability Since e v e n t u a l l y w e shall n e e d to consider the limit o f infinite lattice size, N - - ~ ~ , w e m u s t specify the relation b e t w e e n t and N. Interesting behavior, like fractality, has a c h a n c e of occurring o n l y if the "time" t scales in such a w a y that the total number of white sites W(t) is m u c h less than N. H e n c e a p r e l i m i n a r y q u e s t i o n to be a n s w e r e d is h o w t has to d e p e n d on N in order that this be the case. W e shall do so by first studying the f o l l o w i n g t w o questions.
Question 1. W h a t is the t i m e ~'u at which the last site is visited? W e shall call 'rN the lattice covering time; it is a stochastic variable described by s o m e (in g e n e r a l u n k n o w n ) probability distribution. Question 2. W h a t is the probability LN(x ) that the last site is x (and not s o m e o t h e r site x ' ) ? W e shall refer t o LN(X ) as the last-site probability. 2.1. Mean field and dimension d = 1 It is instructive first to study a m e a n field or " d = ~ " m o d e l [1]. A w a l k e r starts at the origin, x = 0 , of a lattice of N sites, and m a k e s a j u m p to a r a n d o m l y s e l e c t e d site out of the N - 1 u n o c c u p i e d o n e s each unit of time. Let
M . J . A . M . Brummelhuis, H.J. Hilhorst / H o w a random walk covers a finite lattice
37
P(W, t) be the probability that after t jumps there are still W whites sites left. It is easy to write down and solve a recursion in time for this quantity; the result is W(t)
=--~ WP(W,
t) = N e -'IN
as N----> oo.
(1)
W
T h e last site is visited when W(t)= ~7(1), which happens when t = N log[N/ G ( 1 ) ] - N log N. Upon identifying this time with the average lattice covering time ~N we see that
~N=NlogN
for
d=~,
N---~.
(2)
This identification rests upon the replacement of the average of the time evolution by the time evolution of the average. In the mean field case it can be justified by an independent calculation of ~N; in general, it should be correct when P(W, t) is sharply peaked in W, a property that we shall investigate below. Since all sites in the mean field model are equivalent, we have trivially that 1
LN(X ) =
N-1
for x # 0
(3)
(mean field). 0
for x = O
The mathematician Aldous [2] showed in 1983 that on hypercubic lattices with periodic boundary conditions
7rN=AdNIogN
for
d=3,4,5 ....
and
N---~o0.
(4)
This work provides expressions for A d in terms of the probability of return to the origin, and finally leads to [1] A 4 =
1.239
....
A 3 = 1.516 . . . .
(5)
These values were confirmed in computer simulations by Nemirovsky, Mfirtin and Coutinho-Filho [3], who fit their data on two-dimensional lattices to
~s = A2Nlog 2N ,
d = 2,
N----~~ ,
(6)
wit an estimated A 2 = 0.33. It is amusing also to consider the one-dimensional case, where the periodic
38
M.J.A.M.
B r u m m e l h u i s , H . J . Hilhorst / H o w a random walk covers a finite lattice
lattice is a closed loop of N sites. It was shown by Yokoi, Hernfindez-Machado and Ramirez-Piscina [4] that "~U= ½N(N- 1) for d = 1. (Interesting results for the average time needed to visit all sites of a loop at least k times come from simulations by Mirasso and Mfirtin [5].) Subsequently Brummelhuis and Hilhorst found that the last-site probability distribution LN(x ) in d = 1 is given by exactly the same expression, eq. (3), as in the mean field case, a result which at first sight may be somewhat counterintuitive!
2.2. Dimension d >!2 One might now guess that LN(x ) is given by eq. (3) in all dimensions 1 ~< d <~ 2. It is not obvious, however, how to determine LN(X ) for arbitrary dimension by the standard methods of random walk theory. The reason is that " x being the last site visited" is a property that refers to the walker's whereabouts during its whole history. There is, in particular, no simple differential equation from which LN(x ) may be solved. One can of course write LN(x ) in terms of a path integral on all walks, but the resulting expression remains formal for 1 < d < 2. We shall now show how LN(x ) can be obtained, for all d ~> 2, by a method that is based on a separation of time scales. In the course of our argument we shall also prove that the two-dimensional lattice covering time is indeed given by eq. (6), and that A 2 = ,rr
1=0.318 . . . . .
(7)
Let Wt(x ) be the variable that equals 1 if at time t site x is still white, and 0 otherwise. Its average W,(x) is the probability that at time t site x is still white. Let furthermore F,(x) be the first passage time distribution for site x, i.e. the probability that the first visit of site x will take place exactly at time ~-. Then
(8)
W,(x) = 1 - ~ F~(x). ":,"= 0
Standard methods of random walk theory express F,(x) in terms of the random walk G r e e n function. We refer to ref. [1] for the relevant formulas for a hypercubic lattice with periodic boundary conditions. Upon substituting these known results in eq. (8) one arrives at
1
Ca 2 ) e -'/AdN ]x-~-_
for d > 2
(9a)
for d = 2 ,
(9b)
W,(x) =
2 ~log Ixl e - ,,N,ogN
39
M.J.A.M. Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
where C d is a known constant. Both eqs. (9) are valid for N---> o0 and Ixl l, with t at least of order N in (9a) and at least of order N log N in (9b). For x near the periodic boundary of the lattice there are corrections [1]. We first note that by summing eq. (9) on all x one obtains the average total n u m b e r of white sites still left at time t, that is, the quantity W ( t ) discussed before. If one looks for the average lattice covering time ~N by setting this n u m b e r equal to t?(1) and solving to leading order in N, one finds that in dimensions above two, eq. (9a) leads to (4) and that for dimension two eq. (9b) leads to (6) together with (7). Hence for d = 2 we have confirmed the conjecture (6) by Nemirovsky et al. [3] and calculated the constant A 2. This completes the answer to question 1. More detailed information comes from the prefactors in eq. (9). These show that at any given time different sites have different probabilities of still being white, but that these probabilities have time-independent ratios for t of order N or larger, when d > 2, and for t of order N log N or larger, when d = 2. We claim now that the prefactors in (9), when multiplied by the overall normalization factor N - i , are equal to the last-site probabilities LN(X ). The justification is as follows. For large times (t increasing faster than N 2/d) the probability distribution of the random walker position is uniform over the lattice. Therefore all sites which are still white at such times have the same rate of decay to the black state. This explains the x - i n d e p e n d e n c e of the exponentials in eq. (9). That the prefactors in (9) do depend on x is therefore due to a transient effect, namely the fact that the walker starts at t = 0 at the precise site x = 0. H e n c e , in addition to the uniform decay discussed above, sites near the origin have an extra probability of being colored black by the random walker on its initial path away from the origin. Indeed one observes that L N ( x ) is depressed near the origin, and approaches the value N -1 for large x from below. This completes the answer to question 2. 3. Fluctuations on time scale 7N
We now introduce a new time variable a and consider times t such that t = o~N .
(10)
U p o n substituting (10) in (9), using (4) and (6), and summing on x (which just amounts to replacing the prefactor by N), one finds that the total number of white sites still left for t = a~ N and N large is W(a~N) = N 1-~ .
(11)
40
M.J.A.M.
B r u m m e l h u i s , H . J . Hilhorst / H o w a random walk covers a finite lattice
For 0 < a < 1 and N--* ~ this number is much less than the initial number, N, but still much larger than 1. For l < a one should interpret eq. (11) as indicating that with a large probability the lattice is fully black, but that with a small probability some white sites are left. We shall want to know first how sharply the distribution P(W, t) is peaked. We therefore ask the following question.
Question 3. What are the f l u c t u a t i o n s AW2(ti for times t OliN? We remark that if each site were colored white independently with probability N -s, one would have A W 2 1 / 2 = N ~1-~)/2= lye 1/2. Secondly, we would like to know something about the clustering properties of the white sites. We therefore ask the following question, which is a test for fractality. =
Question 4. What is the number Wt(RIx ) of white sites within a distance R from a given white site x? Questions 3 and 4 both involve the correlation between pairs of sites. The key role in answering them is played by the correlation function
AW,(x, y) =- W,(x) Wt(y ) - W,(x) W , ( y ) .
(12)
By a relation which generalizes eq. (8), one can express Wt(x)W,(y) in terms of the probability F~(x, y) that the first visit of the random walker to the set of two sites {x, y} takes place exactly at time r. Since F~(x, y) can again be expressed, by standard methods, in terms of the Green function of the random walk, one obtains an expression for AW,(x, y) on the same footing as eq. (9).
3.1. Dimension d > 2 The properties of AW,(x, y) relevant for our purpose are, for N---~ ~ and a fixed [1],
AW,(x, y) = W~(x) ,
d >12, (13a)
AWt(x, y) = N-2~[cl log N/Ix - yl a-2 + c 2 log 2 N/lx - y12d-4],
d >2. (13b)
(In (13b) we suppose x and y on the scale N l/d of the linear lattice size. Terms with x and y at a fixed distance # 0 are proportional to powers of N less than - a and need no special consideration. See ref. [1].) By summing AW,(x, y) on x and y one obtains the mean square fluctuations
M . J . A . M . Brummelhuis, H.J. Hilhorst / H o w a random walk covers a finite lattice
41
AW2(t). In this summation the terms with x = y and those with I x - Yl large compete for the dominant role. The leading order term in (13b) vanishes exactly [1] when one sums it on x - y and one takes into account the finite size and s m a l l - I x - y[ effects; it is therefore not of interest here. Fig. 2 helps to represent the results for the mean square fluctuations. One finds, also using eq. (9a) for W,(x), that the a d - p l a n e is divided into two regions by the line a = 4/d - 1. Explicitly, AW-(OtTN)I/2=N (1- a ) / 2 AW2(oITFN)I/2
oc N 2 / d - a
~,/1/2
=
for
a>4/d-1,
d>2,
(14a)
log N = (1 - a) -1 1~~2/d-")/cl-~) log 15' for
a<4/d-1,
d>2.
(14b)
T h e fluctuations of eq. (14a) are as for independently colored sites. The calculation actually shows that it stems solely from the x = y terms (eq. (13a)) in the summation, the x ~ y terms of eq. (13b) constituting a negligible correction. Above the critical dimension d = 4 the correlations (13b) are not integrable at the origin and the result (14a) holds [1] for all times a > 0 , one may show that it also holds for the d -- ~ model discussed above. In eq. (14b), since d > 2 , the power of W is larger than ½ but less than 1. This strong enhancement of the fluctuations AW2(t) is due to the dominance of correlations (eq. (13b)) between the most distant pairs of sites x and y. The interpretation is as follows. Initially, that is, for not too large a, the white regions in the lattice are large and the fluctuations AW2(t) are dominated by the fluctuations of their sizes. As time goes on these large regions are broken
'I
d
4~
MF
3
I ,
,
0 Fig. 2. Representation of the properties of the set of white sites in the time(a)-dimension(d) plane. In the region marked LR the fluctuations in the total number of white sites are anomalously large due to long-range pair correlations; in the one marked MF the white sites are essentially randomly distributed, as in mean field. At d = 2 the line of critical points expands into a critical interval along which the exponent fl varies from ½ to 0.
42
M.J.A.M. Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
up into smaller ones, and beyond a critical time a¢ = 4 / d - 1, the fluctuations AW2(t) are dominated by the fluctuations in the n u m b e r of individual white sites. C o m m o n to both regions of eq. (14) is that for a < 1 one has AWa(t)l/2/ W(t)---~O as N - - ~ with t = a~ N, so that the distribution P ( W , t) is sharply p e a k e d for all d > 2. This answers question 3 for d > 2. 3.2. D i m e n s i o n d = 2
F r o m eq. (13b) it is clear that the lower critical dimension d = 2 must be special. Instead of eq. (13b) we now have [1], in addition to eq. (13a), which remains valid,
I cU-Olx - Yl o
AW,(x,y)=[N_Zf( )
for [ x - y [ ~ 0 and fixed as N---~o~,
for
N-'/Zlx-y I fixed
as N---~oo.
(15a)
(15b)
T h e constant c and the function f are known; in particular [1], f ( ¢ ) oc ~ -4~
for ~:~0
(16)
and f ( ~ ) remains finite for ~ of order 1. One sees that for a < ½ expression (15b) can be s u m m e d on all x and y, and that all distances Ix - y[ on the scale contribute. The result is AW2(oFrN)
1/2 ~-
C(ot) N 1-'~ = C(ol) W ( t ) ,
0 < a < ½,
d = 2.
(17)
It shows that the width of P(W, t) is of the same order in N as the average W ( t ) itself. C ( a ) is a known constant which is easily shown to diverge as 1/( ½ - a)l/2 as a l '1 . For ½ < a but still a < 2 , one sees from (16) that eq. (15b) is no longer integrable at the origin, but eq. (15a) still is. H e n c e the behavior of A W , ( x , y) for 1 ~ Ix -- y[ ~ N 1/2 has to be analyzed m o r e precisely. This is done in ref. [1] and we state the result. T h e r e appears a time regime ½ < a < 2, represented by the heavy solid line in fig. 2, where the main contribution to W2(t) comes from pairs of sites with Ix - y [ on the scale N m~), in which ~(a) = 1 -V~7-2.
(18)
As a increases from ½ to 2, this spatial scale shrinks continuously from the linear lattice size N ~/2 to the lattice distance 1. The explicit result for the rms
M.J.A.M. Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
43
fluctuations in this time regime is A w e ( o z r u ) '/2 oc (log N) 1/4 N 3/2-V-~ ,
½
d=2,
(19)
and hence these are much larger than its average 1~' itself. For a = 2 the p o w e r of N in eq. (19) coincides with the one of the mean-field result (14a). This result continues to hold for ot > 2 (note however the interpretation of a > 1 given above), and is solely due to the x = y terms (eq. (13a)) in the summation. We have hereby completed the answer to question 3. We conclude that for d = 2 the distribution P ( W , t) is broad and nontrivial, and that different intervals of a must be distinguished. It remains unexplained that in spite of this broadness we should have been able to determine fN in good agreement with the c o m p u t e r simulations [3] by the argument following eq. (9).
4. Fractality Finally we address question 4 about the occurrence of fractality. The average n u m b e r of white sites within a distance R from a site x known to be white is
W,(Rlx) =
W t - ~ -1 ~ W t ( x ) W t ( x -~- r) r
= N - " K a R a + N " ~ AW~(x, x + r ) , r
(20)
where in the second step we have used (12) and chosen x on scale N ~/a, so that the prefactors from the initial effect can be ignored, and where K a is the v o l u m e of the d-dimensional unit sphere. With the aid of eq. (13b) it is easily checked that for all dimensions d > 2 the first t e r m in (20) dominates the second one as R ~ ~. H e n c e for d > 2 the white sites are statistically u n i f o r m l y distributed at large distances. For d = 2 the situation is different. For times 0 < a < 2 and R independent of N we can use (15b) in (20) and find that
W,(Rlx) ~ R 2-~
for R l a r g e ,
d = 2.
(21)
H e n c e in dimension d = 2 the white sites constitute a f r a c t a l of dimensionality dF=2D
o~
= 2 - ~ r t / N log 2 N
for t, N----~~ and a fixed.
(22)
44
M . J . A . M . Brummelhuis, H.J. Hilhorst / How a random walk covers a finite lattice
In ref. [1] it is shown that for ½ < et < 2 this fractal is essentially confined to a region of linear size N ~ ) , with f l ( a ) given by (18). T h e s e results p r o v i d e an a n s w e r to question 4. W e c o n c l u d e by e m p h a s i z i n g that for d = 2 there is a basic question which has b e e n c i r c u m v e n t e d and left u n a n s w e r e d in this work. We state it explicitly. Q u e s t i o n 5. W h a t is, in d i m e n s i o n d = 2, the distribution P(W, t) of the total n u m b e r W of white sites at time t? T h e interesting time regime is the one of eqs. (10), (7), and (6).
A n a n s w e r to this question would, a m o n g o t h e r things, explain o u r a p p a r e n t success in finding the m e a n lattice covering time ~ by an a p p r o x i m a t e argument.
References [1] [2] [3] [4] [5]
M.J.A.M. Brummelhuis and H.J. Hilhorst, Physica A 176 (1991) 387. D.J. Aldous, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 62 (1983) 361. A.M. Nemirovsky, H.O. Mfirtin and M.D. Coutinho-Filho, Phys. Rev. A 41 (1990) 761. C.S.O. Yokoi, A. Hernfindez-Machado and L. Ramirez-Piscina, Phys. Lett. A 145 (1990) 82. C.R. Mirasso and H.O. M~irtin, Z. Phys. B 82 (1991) 433.