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On returns to the starting site in lattice random walks
Physica 134A (1986) 44w57 North-Holland, Amsterdam
ON RETURNS TO THE STARTING SITE IN LATTICE Barry
RANDOM WALKS
D. HUGHES*
Department of Mathematics, Faculty of Military Studies, University of New South Wales, Duntroon ACT 2600, Australia and Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra ACT26&I, Australia
Received 2 January 1985 Revised 19 July 1985
In sufficiently low-dimensional systems, the conditional mean time to return to the starting site (conditional upon return eventually occurring) is infinite. We examine the conditional mean time 7. to return in a walk of finite duration n steps. For walks of Wlya type, T. is found asymptotically proportional to d/n, n/lo2 n, d/n and log n in dimensions 1,2,3 and 4 respectively. Results are also given for walks with long-ranged transitions, and for a one-dimensional walk in a central potential.
1. Introduction “Fluctuation effects” in lattice random walks have dramatic consequences in walk, with only low dimensions. Pblya’) established that for an unbiased nearest-neighbour stepping on a simple hypercubic lattice of dimension d, return to the starting site is certain if d = 1 or d = 2, while there is a finite probability of escape if d 2 3. Even though return is certain when d = 1 or 2, the
mean
time
d-dimensional
to return lattice
P, = probability
is infinite.
Zd of points
of occupancy
For
an arbitrary
with integer of the starting
random
coordinates,
walk
on the
if we write
site on the nth step
and F, = probability the
theory
of first returning
of recurrent
events*)
to the starting leads
site on the nth step,
to a simple
relation3)
between
the
generating functions
P(5) =
2 P,,t”
m
and F(5) =
n=O
* Present address: 3052, Australia.
c F&T’, II=1
Department
of Mathematics,
University
0378-4371/86/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
of Melbourne,
B.V.
Parkville,
Victoria
444
B.D. HUGHES
viz.
F(5) = 1- P(5))’ Since the return
probability
il F, = F(l-) divergence be certain.
3
of P(5) as 5 -+ l- is a necessary and sufficient condition for return When return is certain, the mean time to return is simply
to
P’(5)
c nF, = F’(l-)
= lim c-1-
n=l
When
is given by
Pm2
.
return
is not certain, the conditional probability of a first return to the occurring on the nth step is F,/R, so the conditional mean recurrence time (the mean time for a first return, considering only those walks which do return) is starting
site
7 = 2 nF,/R = F’(l-)/R
= R-‘(l-
II=1
R)2tiy +
P’(t).
When T < 00, the random walk is called “strongly transient”, and for Polya’s original problem4), the walk is strongly transient if d 2 5. (Strongly transient walks have a number
of simple
properties4).)
How may one investigate dimensionality effects in walks which are not strongly transient? One possibility is a careful examination of walks of large but finite duration, to probe the nature of the ‘divergence’ of 7. Let us write
R, = probability
of a return
to the starting
site occurring
in the first n steps
of the walk and define T, = conditional
mean
so that
r, = i jF;.IR, . j=l
recurrence
time for a walk of n steps
RETURNS TO STARTING SITE IN RANDOM WALKS
44.5
If we introduce the generating function
we see that Q(r) =
Ci
j+J” =
n=l j=l = (1 -
CjJ$$ji j=l
t”-’
n=j
@‘[F’(I).
When the asymptotic behavior of F’(t) as t+ l- is known, we may deduce the large n behaviour of r,,Rn from the behaviour of Q(r) near 5 = 1 using a Tauberian theorem5): The relations
and cn
L-n
P-‘unY~(P),
n+m,
are equivalent, provided that: the sequence {c,} is (ultimately) monotonic; c, SO; p >O; and the function L is slowly varying, in the sense that L(Ax)/L(x) + 1 as x + 00 for each fixed, positive A. The Tauberian theorem is applicable to Cz=, T,,R,,~", since from the definition of r,, the sequence r,,R, is non-negative, and monotonic increasing. As R, - R as n + 03, the asymptotic form of rn is easily deduced from that of r,R,. In this paper we examine the large n behaviour of r,, for a variety of walks which are not strongly transient. In section 2 we examine walks of Polya type (i.e. unbiased walks, with nearest-neighbour stepping), while in section 3 we examine more general walks, including cases in which the mean-square displacement per step is infinite. To illustrate what may occur for walks lacking translational invariance, we consider in section 4 a one-dimensional walk of Gilli$) in which the probability of stepping left or right varies with position. Our results may be summarized as follows: If we define
log 7”
A = limn-m log n
446
B.D. HUGHES
and R denotes
the probability
of eventual
of the classes of walks considered (i) R=l,
O
(ii) R
O
(iii) R < 1 ,
three
return distinct
to the origin, parameter
we find for each
regimes:
A= 0 .
The third of these cases corresponds to strong transience. At the transition between regimes (i) and (ii) we find 7, - n/L,(n), while at the transition from regime (ii) to regime (iii), 7, - L,(n), with the functions L,(n) and L,(n) divergent as n +m, but slowly varying in the sense of the enunciation of the Tauberian theorem.
2. Walks of Pdya type The generating function P(5) for occupancy of the starting site has been extensively studied for walks of Polya type, in which only nearest-neighbour steps are allowed and all allowed steps have equal probability. We simply quote the results here. In one dimension3), P(r) = (1- ,$2)-“2, whence F(r) = 1- (1- #‘2 and it follows that Q(e)
= (1 -
c)-‘(‘(1
Applying the Tauberian that 7, - (2n/7r)1’2.
- t’)-“’ theorem
For the two-dimensional
- 2m1’2(1- ()m3’2 and noting
P(5) = i log
elliptic
(&)
1.. .
square lattice it can be shown7) that
t*)-‘“(1 - k2t2)-“2 dt
is the usual complete
[+
that r(g) = ir(k)
where
K(k) = j- (1-
as
integral.
Consequently,
{1+ ml - 51N
= idr,
we find
RETURNS TO STARTING SITE IN RANDOM WALKS
447
and we find that
am-
(1 - 5)’
log:(S/[l 61) ’
so that for the square lattice
The calculation may be repeated for the triangular lattice, for which’)
P(5) = g
log (&)
(1 + 6([1- 81))
and we find that 27rn 7n - v/3 log2(12n).
Similarly for the hexagonal or honeycomb lattice’) 3V/3 p(r) = zlog
(&)
U+ Nl-
5‘1)) 9
and so
rn - 3d3 log2 (4n) . We note that as we hold the dimensionality of the lattice constant at two and vary the coordination number z (i.e. the number of possible choices for each step) from t = 3 (honeycomb lattice) to z = 4 (square lattice) to z = 6 (triangular lattice) the conditional mean recurrence time T, increases. We also note that by increasing the dimensionality of the lattice from one to two we increase r,,, but this trend does not continue. For the standard three-dimensional lattices (diamond, z = 4; simple cubic, z = 6; body-centred cubic, .z = 8; face-centred cubic, z = 12; and hexagonal close-packed, z = 12),
448
B.D. HUGHES
with P(1) known exactly%“) and c a lattice-dependent as t* l-, Q(t) a (1 - [)-3” and so TV
a n “’
constant. It follows that
as n + 03
in three dimensions. For the four-dimensional
P’(t) a log(l/[l
-
hypercubic lattice P(1) is finite, while”)
61) as5+ 1
and so T.
a log n
asn+m
in four dimensions. In five and higher dimensions, the walk is strongly transient and 7” approaches a finite value as n + cc).Thus we now have an exhaustive qualitative treatment of T” for Polya walks. The divergence of rn as n + 03 for low-dimensional systems indicates that relatively circuitous paths back to the start have significant statistical weight, and this weighting is in some sense maximal in two dimensions. If we define the random variable T, to be the time to return in a walk which does return in n or fewer steps we have rn = (T,), with the angle brackets denoting the appropriate expectation or averaging operation. What is the variance
of Tn when n is large? If we note that n
af, = c j(j - l)F;.IR, + rn - T’, j=l
we are easily able to show that
5 R,{aZ,-
T,,+ &”
=
(1- 5)-‘5’F”(t)
=
Cl- 5)-Y
n=l
As R,{a~- 7, + ~2 is non-negative theorem is immediately applicable.
P”(5) 2wY* [poz--) fw3
and monotonic
.
increasing, the Tauberian
RETURNS TO STARTING SITE IN RANDOM
WALKS
449
In one dimension and in three dimensions, we find that
as t+ l- and so
In two dimensions, however, as n + CO
c
R,{aZ,- rn +
II=1
~26” a
(1 -
Z)-‘/log’(l/[l- 61)
and so
af a
n2/log2 n,
while in four dimensions as n + 03
5 R,(crZ, - r, + r>[”
a (1- t)-*
n=l
giving
The variance at of T,, like the mean r,,, is maximal in Polya walks at dimensionality 2.
3. Translationally invariant walks In a general translationally invariant walk, let p(l) denote the probability of a displacement 1 occurring on any step and define the structure function or characteristic function A(@) corresponding to p by A(@) =
C
e”‘@p(l) .
A straightforward
Fourier analysis3) enables us to express Pn, and hence its
450
B.D. HUGHES
generating
‘(5)
=
function
P, in terms
1
dd@
p)d
I 1 - .$A(@)
of A:
’
B
where
B denotes
the first Brillouin
zone [-7r, ~1~. Hence
A very wide class of walks give rise to the asymptotic l-A(@)-cl@l’
law
as@-+0,
with c independent of @, and the constant p confined to the interval 0
representations
that P(t)
and P’(t)
can diverge
at
5 = 1 only if 1 - A(@) vanishes sufficiently rapidly at some point inside or on the boundary of the first Brillouin zone B. We shall assume for simplicity here that A(@) = 1 in B or on its boundary example, Polya walks on the body-centred entails
no serious
loss of generality.)
only at @ = 0. (This excludes, and face-centred cubic lattices, Whether
or not
P(t)
and
P’(t)
for but will
diverge (and if so, how they will diverge) is governed by the behaviour of the integrand in a hypersphere centred on the origin. Since the volume element in d-dimensional polar coordinates is Adpd-’ dp (with A, = 2+“2/r(d/2) the surface area of the hypersphere of unit radius and p = I@]) we easily see that P(t) diverges as 5 + l- when 0 < d s F, while P’(r) diverges as [ + l- when 0 < d =s 2~. Moreover, for any 6 > 0,
P(t)- u(l-8, P’(5) - V(l where
- r),
O
RETURNS TO STARTING SITE IN RANDOM
WALKS
451
6
and 6
In the appendix we show that as E + 0’ U(E) a
~~@-l,
O
U(E) a
logWE),
d = p ,
V(E)a E+*,
O
V(E) 0: log(l/c),
d = 2~.
Using the Tauberian
7n a ‘ .
theorem we are able to conclude that as n + m
n/log* n ,
d=p,
n
CL<~<*P>
*-d/p
logn,
,
d=2p.
The results derived above for walks of Polya type are now recognized as illustrations of these more general formulae in which circuitous paths receive their greatest weight if d = p, i.e. in the two cases d = p = 1 and d = ,u = 2. 4. One-dimensional walk in a variable environment Few random walk processes in lattices lacking translational invariance are tractable. The classical exceptions to this rule are periodic lattices with simple boundaries, a finite number of defects, or periodic defects. More recently pseudolattices’6) and fractals17) have been analysed and there is a growing body of work on randomized lattices”). In a remarkable paper, Gillis!) solved a one-dimensional problem in which the probability of a step from site I’ to site 1 is given by
B.D. HUGHES
452
P(l
;v,,,+6-J)
Iu = i
where
-1~
l’=O,
;u- E/1’)&1’+,+ ;u+EIE’IS,, I’-,
E -=c1. For E >O there
while
if E
from
the origin
walk. He found
is a bias towards
is a bias away from
of coordinates,
I’ZO,
,
which
the origin
of coordinates,
it. The bias decreases
Gillis
took
as the starting
with distance point
of the
that
where
a (u) (b) n *FI(a, 6; c; t) = c n z n=O (c),n! [with (a), = T(a + n)/f(a)] d enotes the usual hypergeometric asymptotic behaviour of ,FI(u, b; c; z) near z = 1 is easily transformation
function. The found from the
formulae”)
*FI(u, b; c; z) =
T(c)T(c - a - b) T(c - u)r(c - b) + (1 - z)c-=-b
*F,(u, b; a + b - c + 1; 1 - z)
f(c)f(u
+ b - c)
,F,(u,b;c-a-b+l;l-z)
T(a)W) (valid for c - a - b nonintegral)
*F&u, b; a + b; z) =
and
T(u+ b) cm (u),(b), Uu)r(b) n=O (n!)’ x [2$(n + 1) - t,h(n + a) - t,h(n + b) - log(1 - z)](l - 2)“.
[Here $(z) = d/dz log r(z).] The numerator in Gillis’ expression for P(e) diverges as e+ l- when E 2 -i; the denominator diverges as ,$+ l- when E a k. Asymptotic expansions of F(t) and F’(t) valid as ,$+ l- are easily deduced and we find that R=
lal-‘-l,
(
1
)
-lQ&, -+<1,
RETURNS TO STARTING SITE IN RANDOM WALKS ’
3/z+& n
,
n/log2n,
T?Taz
n
m-s
,
453
-lee<-;, E = -i, -;
. logn,
while T”-+2&/(2&-1),
+&Cl.
5. Discussion
Let pn denote the expected number of returns to the starting site in an n-step walk. It is tempting to argue that for large n P”Tll - constant * n ;
we have of course no right to expect that the constant should be unity. Montroll and Weissr9) have shown for translationally invariant walks on periodic lattices that
/4(z)=
c
/4”=
n=O
F(S) (1- 5Nl- F(5)) ’
For transient walks (where the return probability we have
R = F(1) is less than unity),
R ‘(‘)-
(1- R)(l-
5)
and so R A%--
1-R’
We know that T-~approaches transient in the sense defined n + COcannot hold for strongly that it also does not hold in strongly transient.
a constant if and only if the walk is strongly in section 1. Therefore the relation pu,r,, m n as transient walks. The analysis of section 3 shows general for walks which are transient but not
B.D. HUGHES
454
What
happens
for walks
in which
eventual
return
to the
starting
site is
certain? For the class of walks analysed in section 3, characterized by the asymptotic behaviour of the structure function 1 - A(@) m c(@[” as t+ ll, we found
that the walk is recurrent (11 - F(5) = P(t)-’
for dimensions 5)1-&N )
l-. It follows (l-
P(5) Oc
d
m
{log(l/[l as [+
d G CL,with
5]))-’ ,
d =
p,
that
[p2)
(I- 5)-l log(l/[l-
d
d=
P >
and so as n-+a l-d/p n , d
we find that r,,~,, 0~ n for d
mean recurrence the nature of the
individual transitions is subtle. It is therefore not surprising the resistance of a random walk, in which the distribution
that the problem of of lengths of closed
loops is particularly dimension20*21).
in more
important,
is relatively
intractable
than
one
The author is indebted to Drs M. Sahimi, C.E. Wayne and J. Kirkwood discussions of the subtleties of the structure of random walk paths.
for
Acknowledgement
Appendix We briefly derive here leading order functions U(F) and V(E) defined in section
asymptotic representations of the 3 valid as F + O+. The technique we
RETURNS
TO STARTING
SITE IN RANDOM
WALKS
455
employ”) is based upon the Mellin transform. The absolute convergence of the integrals we manipulate below enables us to interchange orders of integration and translate vertical integration lines in the complex plane at will. We define the Mellin transform 0 of U by
for those complex values of s for which the integral is convergent. Inserting the integral defining V, interchanging the orders of integration, and remembering that m
I
s-1 tS-‘@ + t)-’
dt
= =
sin(m)
O
0
we find without difficulty that, so long as u = Re(s) is confined to the strip 1 - d/p < (T < 1 (we are considering only the case 0 < d c p here, so the strip is not void), at-r+d
Af%
,.
v(s) = (27r)d sin(rs) ps - p + d ’ Using the inversion theorem for Mellin transforms, we have the relation o+im
(I(E)=&
j
EVSfi(s)ds,
l-d/pC<=Re(s)
rr-im
The dominant small E behavior of U(E) is given by the residue of the integrand at the singularity at s = 1 - d/p. When d < p this singularity is a simple pole, and we find that U(E)
0:
Edlp-l.
For d = p the singularity is a double pole, and
U(E) o:logwE)
.
The analysis of V(E) is similar and we omit most of the details. To evaluate
456
B.D. HUGHES
the Mellin transform
c(s) in closed form we need only note that
ts-‘(a + t)-’ dt
f-l@ + t)-’ dt - = - ; 0
0
= (1- S)%-cC_2
sin(7rS)
’
0 < Re(s) < 2.
We find that Adc”-2T(l
1
v(S)=
_
s)
(27r)d sin(ns)
g+*)+d
p(s - 2)+ d '
for 2 - d/p < Re(s) < 2. Hence as V(E)
0:
.F-*+dl* )
E +
0'
O
V(E) rx lOg(l/E) , d = 2~.
References 1) G. Polya, Math. Ann. 83 (1921) 149. to Probability 2) W. Feller, An Introduction 3) 4) 5) 6) 7) 8) 9) 10) 11)
12) 13) 14)
Theory and Its Applications, vol. 1, 3rd ed. (Wiley, New York, 1%8). M.N. Barber and B.W. Ninham, Random and Restricted Walks: Theory and Applications (Gordon and Breach, New York, 1970). N. Jain and S. Orey, Israel J. Math. 6 (1968) 373. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. (Wiley, New York, 1971). J. Gillis, Quart. J. Math. (Oxford, 2nd series) 7 (1956) 144. F.S. Henyey and V. Seshadri, J. Chem. Phys. 76 (1982) 5530. G.N. Watson, Quart. J. Math. (Oxford, 1st series) 10 (1939) 266. S. Ishioka and M. Koiwa, Phil. Mag. A37 (1978) 517. E.W. Montroll, J. Sot. Indust. Appl. Math. 4 (1956) 241. B.D. Hughes and S. Prager, in: The Mathematics and Physics of Disordered Media, B.D. Hughes and B.W. Ninham, eds. Lecture Notes in Mathematics, vol. 1035 (Springer, Berlin, 1983), p. 1. K. Lindenberg, V. Seshadri, K.E. Shuler and G.H. Weiss, J. Stat. Phys. 23 (1980) 11. E.W. Montroll and B.J. West, in: Fluctuation Phenomena, E.W. Montroll and J.L. Lebowitz, eds. (North-Holland, Amsterdam, 1979) p. 61. E.W. Montroll and M.F. Shlesinger, in: Nonequilibrium Phenomena II: From Stochastics to
Hydrodynamics, 1. 15) B.D. Hughes,
J.L. Lebowitz E.W. Montroll
and E.W. Montroll, and M.F. Shlesinger,
eds. (North-Holland,
Amsterdam,
J. Stat. Phys. 28 (1982) 111.
1984) p.
RETURNS
TO STARTING
SITE IN RANDOM
WALKS
16) B.D. Hughes, M. Sahimi and H.T. Davis, Physica 1U)A (1983) 515. 17) R. Rammal and G. Toulouse, J. de Phys. Lett. 44 (1983) L13. 18) M. Abramowitz and LA. Stegun 1965, Handbook of Mathematical Functions (Dover, York, 1965), p. 559. 19) E.W. Montroll and G.H. Weiss, J. Math. Phys. 6 (l%S) 167. 20) J.A. Banavar, A.B. Harris and J. Koplik, Phys. Rev. Lett. 51 (1983) 1115. 21) M. Sahimi, G.R. Jerauld, L.E. Striven and H.T. Davis, Phys. Rev. A 29 (1984) 3397. 22) B. Davies, Integral Transforms and Their Applications (Springer, New York, 1978).
457
New
ON RETURNS TO THE STARTING SITE IN LATTICE Barry
RANDOM WALKS
D. HUGHES*
Department of Mathematics, Faculty of Military Studies, University of New South Wales, Duntroon ACT 2600, Australia and Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra ACT26&I, Australia
Received 2 January 1985 Revised 19 July 1985
In sufficiently low-dimensional systems, the conditional mean time to return to the starting site (conditional upon return eventually occurring) is infinite. We examine the conditional mean time 7. to return in a walk of finite duration n steps. For walks of Wlya type, T. is found asymptotically proportional to d/n, n/lo2 n, d/n and log n in dimensions 1,2,3 and 4 respectively. Results are also given for walks with long-ranged transitions, and for a one-dimensional walk in a central potential.
1. Introduction “Fluctuation effects” in lattice random walks have dramatic consequences in walk, with only low dimensions. Pblya’) established that for an unbiased nearest-neighbour stepping on a simple hypercubic lattice of dimension d, return to the starting site is certain if d = 1 or d = 2, while there is a finite probability of escape if d 2 3. Even though return is certain when d = 1 or 2, the
mean
time
d-dimensional
to return lattice
P, = probability
is infinite.
Zd of points
of occupancy
For
an arbitrary
with integer of the starting
random
coordinates,
walk
on the
if we write
site on the nth step
and F, = probability the
theory
of first returning
of recurrent
events*)
to the starting leads
site on the nth step,
to a simple
relation3)
between
the
generating functions
P(5) =
2 P,,t”
m
and F(5) =
n=O
* Present address: 3052, Australia.
c F&T’, II=1
Department
of Mathematics,
University
0378-4371/86/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
of Melbourne,
B.V.
Parkville,
Victoria
444
B.D. HUGHES
viz.
F(5) = 1- P(5))’ Since the return
probability
il F, = F(l-) divergence be certain.
3
of P(5) as 5 -+ l- is a necessary and sufficient condition for return When return is certain, the mean time to return is simply
to
P’(5)
c nF, = F’(l-)
= lim c-1-
n=l
When
is given by
Pm2
.
return
is not certain, the conditional probability of a first return to the occurring on the nth step is F,/R, so the conditional mean recurrence time (the mean time for a first return, considering only those walks which do return) is starting
site
7 = 2 nF,/R = F’(l-)/R
= R-‘(l-
II=1
R)2tiy +
P’(t).
When T < 00, the random walk is called “strongly transient”, and for Polya’s original problem4), the walk is strongly transient if d 2 5. (Strongly transient walks have a number
of simple
properties4).)
How may one investigate dimensionality effects in walks which are not strongly transient? One possibility is a careful examination of walks of large but finite duration, to probe the nature of the ‘divergence’ of 7. Let us write
R, = probability
of a return
to the starting
site occurring
in the first n steps
of the walk and define T, = conditional
mean
so that
r, = i jF;.IR, . j=l
recurrence
time for a walk of n steps
RETURNS TO STARTING SITE IN RANDOM WALKS
44.5
If we introduce the generating function
we see that Q(r) =
Ci
j+J” =
n=l j=l = (1 -
CjJ$$ji j=l
t”-’
n=j
@‘[F’(I).
When the asymptotic behavior of F’(t) as t+ l- is known, we may deduce the large n behaviour of r,,Rn from the behaviour of Q(r) near 5 = 1 using a Tauberian theorem5): The relations
and cn
L-n
P-‘unY~(P),
n+m,
are equivalent, provided that: the sequence {c,} is (ultimately) monotonic; c, SO; p >O; and the function L is slowly varying, in the sense that L(Ax)/L(x) + 1 as x + 00 for each fixed, positive A. The Tauberian theorem is applicable to Cz=, T,,R,,~", since from the definition of r,, the sequence r,,R, is non-negative, and monotonic increasing. As R, - R as n + 03, the asymptotic form of rn is easily deduced from that of r,R,. In this paper we examine the large n behaviour of r,, for a variety of walks which are not strongly transient. In section 2 we examine walks of Polya type (i.e. unbiased walks, with nearest-neighbour stepping), while in section 3 we examine more general walks, including cases in which the mean-square displacement per step is infinite. To illustrate what may occur for walks lacking translational invariance, we consider in section 4 a one-dimensional walk of Gilli$) in which the probability of stepping left or right varies with position. Our results may be summarized as follows: If we define
log 7”
A = limn-m log n
446
B.D. HUGHES
and R denotes
the probability
of eventual
of the classes of walks considered (i) R=l,
O
(ii) R
O
(iii) R < 1 ,
three
return distinct
to the origin, parameter
we find for each
regimes:
A= 0 .
The third of these cases corresponds to strong transience. At the transition between regimes (i) and (ii) we find 7, - n/L,(n), while at the transition from regime (ii) to regime (iii), 7, - L,(n), with the functions L,(n) and L,(n) divergent as n +m, but slowly varying in the sense of the enunciation of the Tauberian theorem.
2. Walks of Pdya type The generating function P(5) for occupancy of the starting site has been extensively studied for walks of Polya type, in which only nearest-neighbour steps are allowed and all allowed steps have equal probability. We simply quote the results here. In one dimension3), P(r) = (1- ,$2)-“2, whence F(r) = 1- (1- #‘2 and it follows that Q(e)
= (1 -
c)-‘(‘(1
Applying the Tauberian that 7, - (2n/7r)1’2.
- t’)-“’ theorem
For the two-dimensional
- 2m1’2(1- ()m3’2 and noting
P(5) = i log
elliptic
(&)
1.. .
square lattice it can be shown7) that
t*)-‘“(1 - k2t2)-“2 dt
is the usual complete
[+
that r(g) = ir(k)
where
K(k) = j- (1-
as
integral.
Consequently,
{1+ ml - 51N
= idr,
we find
RETURNS TO STARTING SITE IN RANDOM WALKS
447
and we find that
am-
(1 - 5)’
log:(S/[l 61) ’
so that for the square lattice
The calculation may be repeated for the triangular lattice, for which’)
P(5) = g
log (&)
(1 + 6([1- 81))
and we find that 27rn 7n - v/3 log2(12n).
Similarly for the hexagonal or honeycomb lattice’) 3V/3 p(r) = zlog
(&)
U+ Nl-
5‘1)) 9
and so
rn - 3d3 log2 (4n) . We note that as we hold the dimensionality of the lattice constant at two and vary the coordination number z (i.e. the number of possible choices for each step) from t = 3 (honeycomb lattice) to z = 4 (square lattice) to z = 6 (triangular lattice) the conditional mean recurrence time T, increases. We also note that by increasing the dimensionality of the lattice from one to two we increase r,,, but this trend does not continue. For the standard three-dimensional lattices (diamond, z = 4; simple cubic, z = 6; body-centred cubic, .z = 8; face-centred cubic, z = 12; and hexagonal close-packed, z = 12),
448
B.D. HUGHES
with P(1) known exactly%“) and c a lattice-dependent as t* l-, Q(t) a (1 - [)-3” and so TV
a n “’
constant. It follows that
as n + 03
in three dimensions. For the four-dimensional
P’(t) a log(l/[l
-
hypercubic lattice P(1) is finite, while”)
61) as5+ 1
and so T.
a log n
asn+m
in four dimensions. In five and higher dimensions, the walk is strongly transient and 7” approaches a finite value as n + cc).Thus we now have an exhaustive qualitative treatment of T” for Polya walks. The divergence of rn as n + 03 for low-dimensional systems indicates that relatively circuitous paths back to the start have significant statistical weight, and this weighting is in some sense maximal in two dimensions. If we define the random variable T, to be the time to return in a walk which does return in n or fewer steps we have rn = (T,), with the angle brackets denoting the appropriate expectation or averaging operation. What is the variance
of Tn when n is large? If we note that n
af, = c j(j - l)F;.IR, + rn - T’, j=l
we are easily able to show that
5 R,{aZ,-
T,,+ &”
=
(1- 5)-‘5’F”(t)
=
Cl- 5)-Y
n=l
As R,{a~- 7, + ~2 is non-negative theorem is immediately applicable.
P”(5) 2wY* [poz--) fw3
and monotonic
.
increasing, the Tauberian
RETURNS TO STARTING SITE IN RANDOM
WALKS
449
In one dimension and in three dimensions, we find that
as t+ l- and so
In two dimensions, however, as n + CO
c
R,{aZ,- rn +
II=1
~26” a
(1 -
Z)-‘/log’(l/[l- 61)
and so
af a
n2/log2 n,
while in four dimensions as n + 03
5 R,(crZ, - r, + r>[”
a (1- t)-*
n=l
giving
The variance at of T,, like the mean r,,, is maximal in Polya walks at dimensionality 2.
3. Translationally invariant walks In a general translationally invariant walk, let p(l) denote the probability of a displacement 1 occurring on any step and define the structure function or characteristic function A(@) corresponding to p by A(@) =
C
e”‘@p(l) .
A straightforward
Fourier analysis3) enables us to express Pn, and hence its
450
B.D. HUGHES
generating
‘(5)
=
function
P, in terms
1
dd@
p)d
I 1 - .$A(@)
of A:
’
B
where
B denotes
the first Brillouin
zone [-7r, ~1~. Hence
A very wide class of walks give rise to the asymptotic l-A(@)-cl@l’
law
as@-+0,
with c independent of @, and the constant p confined to the interval 0
representations
that P(t)
and P’(t)
can diverge
at
5 = 1 only if 1 - A(@) vanishes sufficiently rapidly at some point inside or on the boundary of the first Brillouin zone B. We shall assume for simplicity here that A(@) = 1 in B or on its boundary example, Polya walks on the body-centred entails
no serious
loss of generality.)
only at @ = 0. (This excludes, and face-centred cubic lattices, Whether
or not
P(t)
and
P’(t)
for but will
diverge (and if so, how they will diverge) is governed by the behaviour of the integrand in a hypersphere centred on the origin. Since the volume element in d-dimensional polar coordinates is Adpd-’ dp (with A, = 2+“2/r(d/2) the surface area of the hypersphere of unit radius and p = I@]) we easily see that P(t) diverges as 5 + l- when 0 < d s F, while P’(r) diverges as [ + l- when 0 < d =s 2~. Moreover, for any 6 > 0,
P(t)- u(l-8, P’(5) - V(l where
- r),
O
RETURNS TO STARTING SITE IN RANDOM
WALKS
451
6
and 6
In the appendix we show that as E + 0’ U(E) a
~~@-l,
O
U(E) a
logWE),
d = p ,
V(E)a E+*,
O
V(E) 0: log(l/c),
d = 2~.
Using the Tauberian
7n a ‘ .
theorem we are able to conclude that as n + m
n/log* n ,
d=p,
n
CL<~<*P>
*-d/p
logn,
,
d=2p.
The results derived above for walks of Polya type are now recognized as illustrations of these more general formulae in which circuitous paths receive their greatest weight if d = p, i.e. in the two cases d = p = 1 and d = ,u = 2. 4. One-dimensional walk in a variable environment Few random walk processes in lattices lacking translational invariance are tractable. The classical exceptions to this rule are periodic lattices with simple boundaries, a finite number of defects, or periodic defects. More recently pseudolattices’6) and fractals17) have been analysed and there is a growing body of work on randomized lattices”). In a remarkable paper, Gillis!) solved a one-dimensional problem in which the probability of a step from site I’ to site 1 is given by
B.D. HUGHES
452
P(l
;v,,,+6-J)
Iu = i
where
-1~
l’=O,
;u- E/1’)&1’+,+ ;u+EIE’IS,, I’-,
E -=c1. For E >O there
while
if E
from
the origin
walk. He found
is a bias towards
is a bias away from
of coordinates,
I’ZO,
,
which
the origin
of coordinates,
it. The bias decreases
Gillis
took
as the starting
with distance point
of the
that
where
a (u) (b) n *FI(a, 6; c; t) = c n z n=O (c),n! [with (a), = T(a + n)/f(a)] d enotes the usual hypergeometric asymptotic behaviour of ,FI(u, b; c; z) near z = 1 is easily transformation
function. The found from the
formulae”)
*FI(u, b; c; z) =
T(c)T(c - a - b) T(c - u)r(c - b) + (1 - z)c-=-b
*F,(u, b; a + b - c + 1; 1 - z)
f(c)f(u
+ b - c)
,F,(u,b;c-a-b+l;l-z)
T(a)W) (valid for c - a - b nonintegral)
*F&u, b; a + b; z) =
and
T(u+ b) cm (u),(b), Uu)r(b) n=O (n!)’ x [2$(n + 1) - t,h(n + a) - t,h(n + b) - log(1 - z)](l - 2)“.
[Here $(z) = d/dz log r(z).] The numerator in Gillis’ expression for P(e) diverges as e+ l- when E 2 -i; the denominator diverges as ,$+ l- when E a k. Asymptotic expansions of F(t) and F’(t) valid as ,$+ l- are easily deduced and we find that R=
lal-‘-l,
(
1
)
-lQ&, -+<1,
RETURNS TO STARTING SITE IN RANDOM WALKS ’
3/z+& n
,
n/log2n,
T?Taz
n
m-s
,
453
-lee<-;, E = -i, -;
. logn,
while T”-+2&/(2&-1),
+&Cl.
5. Discussion
Let pn denote the expected number of returns to the starting site in an n-step walk. It is tempting to argue that for large n P”Tll - constant * n ;
we have of course no right to expect that the constant should be unity. Montroll and Weissr9) have shown for translationally invariant walks on periodic lattices that
/4(z)=
c
/4”=
n=O
F(S) (1- 5Nl- F(5)) ’
For transient walks (where the return probability we have
R = F(1) is less than unity),
R ‘(‘)-
(1- R)(l-
5)
and so R A%--
1-R’
We know that T-~approaches transient in the sense defined n + COcannot hold for strongly that it also does not hold in strongly transient.
a constant if and only if the walk is strongly in section 1. Therefore the relation pu,r,, m n as transient walks. The analysis of section 3 shows general for walks which are transient but not
B.D. HUGHES
454
What
happens
for walks
in which
eventual
return
to the
starting
site is
certain? For the class of walks analysed in section 3, characterized by the asymptotic behaviour of the structure function 1 - A(@) m c(@[” as t+ ll, we found
that the walk is recurrent (11 - F(5) = P(t)-’
for dimensions 5)1-&N )
l-. It follows (l-
P(5) Oc
d
m
{log(l/[l as [+
d G CL,with
5]))-’ ,
d =
p,
that
[p2)
(I- 5)-l log(l/[l-
d
d=
P >
and so as n-+a l-d/p n , d
we find that r,,~,, 0~ n for d
mean recurrence the nature of the
individual transitions is subtle. It is therefore not surprising the resistance of a random walk, in which the distribution
that the problem of of lengths of closed
loops is particularly dimension20*21).
in more
important,
is relatively
intractable
than
one
The author is indebted to Drs M. Sahimi, C.E. Wayne and J. Kirkwood discussions of the subtleties of the structure of random walk paths.
for
Acknowledgement
Appendix We briefly derive here leading order functions U(F) and V(E) defined in section
asymptotic representations of the 3 valid as F + O+. The technique we
RETURNS
TO STARTING
SITE IN RANDOM
WALKS
455
employ”) is based upon the Mellin transform. The absolute convergence of the integrals we manipulate below enables us to interchange orders of integration and translate vertical integration lines in the complex plane at will. We define the Mellin transform 0 of U by
for those complex values of s for which the integral is convergent. Inserting the integral defining V, interchanging the orders of integration, and remembering that m
I
s-1 tS-‘@ + t)-’
dt
= =
sin(m)
O
0
we find without difficulty that, so long as u = Re(s) is confined to the strip 1 - d/p < (T < 1 (we are considering only the case 0 < d c p here, so the strip is not void), at-r+d
Af%
,.
v(s) = (27r)d sin(rs) ps - p + d ’ Using the inversion theorem for Mellin transforms, we have the relation o+im
(I(E)=&
j
EVSfi(s)ds,
l-d/pC<=Re(s)
rr-im
The dominant small E behavior of U(E) is given by the residue of the integrand at the singularity at s = 1 - d/p. When d < p this singularity is a simple pole, and we find that U(E)
0:
Edlp-l.
For d = p the singularity is a double pole, and
U(E) o:logwE)
.
The analysis of V(E) is similar and we omit most of the details. To evaluate
456
B.D. HUGHES
the Mellin transform
c(s) in closed form we need only note that
ts-‘(a + t)-’ dt
f-l@ + t)-’ dt - = - ; 0
0
= (1- S)%-cC_2
sin(7rS)
’
0 < Re(s) < 2.
We find that Adc”-2T(l
1
v(S)=
_
s)
(27r)d sin(ns)
g+*)+d
p(s - 2)+ d '
for 2 - d/p < Re(s) < 2. Hence as V(E)
0:
.F-*+dl* )
E +
0'
O
V(E) rx lOg(l/E) , d = 2~.
References 1) G. Polya, Math. Ann. 83 (1921) 149. to Probability 2) W. Feller, An Introduction 3) 4) 5) 6) 7) 8) 9) 10) 11)
12) 13) 14)
Theory and Its Applications, vol. 1, 3rd ed. (Wiley, New York, 1%8). M.N. Barber and B.W. Ninham, Random and Restricted Walks: Theory and Applications (Gordon and Breach, New York, 1970). N. Jain and S. Orey, Israel J. Math. 6 (1968) 373. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. (Wiley, New York, 1971). J. Gillis, Quart. J. Math. (Oxford, 2nd series) 7 (1956) 144. F.S. Henyey and V. Seshadri, J. Chem. Phys. 76 (1982) 5530. G.N. Watson, Quart. J. Math. (Oxford, 1st series) 10 (1939) 266. S. Ishioka and M. Koiwa, Phil. Mag. A37 (1978) 517. E.W. Montroll, J. Sot. Indust. Appl. Math. 4 (1956) 241. B.D. Hughes and S. Prager, in: The Mathematics and Physics of Disordered Media, B.D. Hughes and B.W. Ninham, eds. Lecture Notes in Mathematics, vol. 1035 (Springer, Berlin, 1983), p. 1. K. Lindenberg, V. Seshadri, K.E. Shuler and G.H. Weiss, J. Stat. Phys. 23 (1980) 11. E.W. Montroll and B.J. West, in: Fluctuation Phenomena, E.W. Montroll and J.L. Lebowitz, eds. (North-Holland, Amsterdam, 1979) p. 61. E.W. Montroll and M.F. Shlesinger, in: Nonequilibrium Phenomena II: From Stochastics to
Hydrodynamics, 1. 15) B.D. Hughes,
J.L. Lebowitz E.W. Montroll
and E.W. Montroll, and M.F. Shlesinger,
eds. (North-Holland,
Amsterdam,
J. Stat. Phys. 28 (1982) 111.
1984) p.
RETURNS
TO STARTING
SITE IN RANDOM
WALKS
16) B.D. Hughes, M. Sahimi and H.T. Davis, Physica 1U)A (1983) 515. 17) R. Rammal and G. Toulouse, J. de Phys. Lett. 44 (1983) L13. 18) M. Abramowitz and LA. Stegun 1965, Handbook of Mathematical Functions (Dover, York, 1965), p. 559. 19) E.W. Montroll and G.H. Weiss, J. Math. Phys. 6 (l%S) 167. 20) J.A. Banavar, A.B. Harris and J. Koplik, Phys. Rev. Lett. 51 (1983) 1115. 21) M. Sahimi, G.R. Jerauld, L.E. Striven and H.T. Davis, Phys. Rev. A 29 (1984) 3397. 22) B. Davies, Integral Transforms and Their Applications (Springer, New York, 1978).
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