*
Notes taken by Mr. Michael Shelor from a series of lectures delivered
at the Summer Study Institute on Combinatorial Mathematics and Its Applications in the Natural Sciences, held on June 2-27, 1969, in Chapel Hill,
and sponsored by the U.S. Air Force Office of Scientific Research under
Air Force Grant No. AFOSR-68-1406 and the U.S. Army Research Office
(Durham) under Grant No. DA-ARO-D-31-124-G910.
0
6
5
4
7
1
0
6
8
7
2
1
9
8
7
3
1
9
8
7
0 6
5 4 9 8 7 1 2 3
7
8
9
1
3
5
2
4
6
0
2
7
8
9
1
7
8
9
2
4
3
6
1
3
7
8
3 9
4 5
5 6 0
0 1 2
5
0
2
4
7
8
9
6
1
3
5
7
6 0
1 2
3 4
8
9
0
2
4
6
1
3
2 3
3 4
4 5
5 6
6 0
0 1
7 8
8 9
5 9 7
4
5
6
0
1
2
9
7
8
3
2
9
8
5 2
4 3
3 4
9 5
8 6
7 0
6 1
0 7
1 9
2 8
4 6
5 0
6
0
1
2
3
8
7
9
1
2
3
4
5
9
8
7
COM BIN A TOR I A L
MAT HEM A TIC S
YEA R
February 1969 - June 1970
RANDOM WALKS ON LATTICES
by
Elliott W. Montro11 *
University of Rochester, Rochester, New York
University of North Carolina at Chapel Hill
Department of Statistics
Institute of Statistics Mimeo Series No. 600.22
FEBRUARY 1970
Random Walks on Lattices
by
Elliott W. Montroll *
University of Rochester
Rochester, New York
1.
INTRODUCTION
TYPICAL PROBLEMS
We shall be concerned principally with counting the number of ways
certain events can occur on lattices, where, by a lattice, we mean a periodic collection of points.
These problems have importance to many branches
of chemistry and solid-state physics since the lattice may be thought of
as a mathematical model of a crystal, with the points of the lattice corresponding to atoms of the crystal.
The most important problem to be dealt with is that of random walks
on periodic space lattices.
The random walk problem will be emphasized
here because, as we shall see, many of the features of this problem also
arise in other lattice problems.
The random walk problem on a crystal
lattice arises in many different physical situations.
For example, sup-
pose the crystal lattice has an interstitial defect, i.e., there is one
atom which is a little different from its neighbors and, thus, may move
from one point to another.
In this case, the diffusion constant for im-
purities in the crystal depend on the random walk in lattices.
For
another example, consider the process of photosynthesis, where energy from
*,
Notes taken by Mr. Michael Shelor from a series of lectures delivered
at the Summer Study Institute on Combinatorial Mathematics and Its Applications in the Natural Sciences, held on June 2-27, 1969, in Chapel Hill,
and sponsored by the U.S. Air Force Office of Scientific Research under
Air Force Grant No. AFOSR-68-l406 and the U.S. Army Research Office
(Durham) under Grant No. DA-ARO-D-3l-l24-G9lO.
2
a light source causes a molecule to go into an "excited" state, and this
excitation is passed from molecule to molecule until it reaches a "trap"
where the important features of the process start.
In this case, the ex-
istence of a molecule in an "excited" state corresponds to the random
walker.
Another type of situation involving random walks is the annealing
of metal alloys.
While the alloy is hot, the structure is porous, but as
it cools it assumes a crystalline lattice structure.
In doing so, how-
ever, pome of the atoms assume the wrong positions and, consequently, they
must move into their correct positions.
In this situation, as the atoms
move to form an equilibrium state, the important question is the number of
new positions occupied by an atom during a certain number of steps.
A second important lattice statistics problem is that involving dimers,
or diatomic molecules.
Here we consider the situation in which dimers are
absorbed on the surface of a lattice, each dimer thereby occupying two
adjacent lattice sites.
We are interested here in the number of different
possible configurations of dimers on the lattice surface.
Another lattice statistics problem, one which appears in the theory
of phase transitions, is the so-called Ising problem (see [3], p.121).
The Ising problem turns out to be equivalent not only to the dimer problem
but also to the mathematical problem of counting the number of ways of
drawing closed graphs on a lattice which has a specific number of fixed
bonds.
The problem of vibration in chemical lattices, which at first glance
does not seem to have much combinatorial significance, actually turns out
to be very similar to the random walk problem.
A final problem is that of the motion of electrons in a lattice.
As
a model of the crystal in this case, we could use nodes (atoms) with wires
3
connecting them so that we can think of the electrons as traveling along
the wires.
This problem, too, is very similar to the random walk problem .
. Thus, the above-mentioned problems are some of the more important lattice statistics problems.
Before proceeding to discuss the most important
of these, the random walk problem, we give a formal definition of a 3D
(i.e., 3 dimensional) lattice.
vectors
i ,i ,
l 2
and
i .
3
Consider a set of three non-coplanar unit
The endpoints of the vectors
s
which are generated by letting the
±2, •.•
SIS
range through the integers
form a space lattice of three dimensions.
0, ±l,
Periodic lattices of
arbitrary dimensionality can be defined in a similar manner.
2.
RANDOM WALKS ON LATTICES
THE POLYA PROBLEM
There will be three general types of questions which we shall be interested in concerning random walks on lattices:
1)
What is the probability that the random walker travels from a
starting point
a
to a point
b
in
n
steps?
If
~ =~,
then we have
the classical Polya problem, i.e., to determine the probability that the
walker returns to his starting point.
2)
The first passage time problem, i.e., what is the probability that
a walker arrives at a point
3)
s
for the first time after
n
steps?
What are the number of distinct lattice points visited by the ran-
dom walker after
n
steps?
4
The present section will be devoted primarily to answering the first
of these three questions.
We consider first the case of a finite
odic boundary conditions.
points (where
d
lattice with peri-
NxNx ••• xN
That is, the lattice will have
d
N
lattice
is the dimensionality) and it will be a torus.
In the
lD case, this would correspond to a ring of regularly spaced points. In the
2D case, it corresponds to a square grid of points wrapped into the form of
a torus, etc.
Define
p (s)
t -
F (s)
t -
=
prob. that walker is at
s
after
prob. that walker is at
s
for first time after
t
t
steps
steps.
Without loss of generality, we can assume that the walker starts at the
origin.
Thus
Po(~)
6
O,~
{~
0
if
s
if
s .; O.
Further, we assume that the walker takes steps at regular time intervals,
i.e.,
t
=
0,1,2, . . . .
With our periodic boundary conditions, it follows that, letting
s
etc.
Finally we make the assumption that the random walk is a stationary random
process, i.e., the transition probability that the walker goes from
Sl
to
5
s
at a given time
t
is independent of
pds'
at time
-+ S
t,
i.e.,
=
t}
p(E.--E.-')
where we make the additional assumption that this probability depends only
on the distance vector
s-s'.
Under the above assumptions, it follows that the random walk constitutes a Markov chain and thus all properties of Markov chains hold.
Furthermore, it is obvious that
(1)
and
1.
We now see that we
~ave
(2)
the difference equation
(3)
P (s)
t -
Define the generating fUijction
t
z p (s)
t -
t:20
then, by substituting the difference equation in (4), we have
Z
L
s'
Z
L
s'
( 4)
6
which implies
p (~, z) -
Z
L
p (~-~ , )P (~' ,z)
<5
s'
(5)
~,Q
This last is a Green's function type of equation.
A simple expression for
all
s
LsP(~,z)
follows from (5) by summing over
and recalling (2):
L P(~,z)
L
+ z
1
p (~-~ , )P (~' , z)
~,~'
s
+ z
1
L
p (~")
L
p (~")
s"
+ z
1
s"
=
1
+ z
L P (~-~" ,z) ;
[~"
~-~']
s
L p (~, z)
s
L P(~,z)
s
so that
1
L P(~,z)
(6)
l-z
s
Another function which is of importance in this study is the finite
Fourier transform
L P(~,z)
e
2TIir-s/N
--
(7)
s
where
r
1
~
r.
1
~
N-L
This is just as good as a function to discuss as the generating function,
since, by taking the Fourier inverse, we can get the generating function
7
from
u(z,(2TIE/N».
Further, let us denote the finite Fourier transform
of the transition probabilities by
L p(~)
e
2TIir·s/N
--
(8)
.
s
This is called the struature funation.
u -
It follows that
ZUA
I
and hence
1
u
(9)
l-zA •
Notice that the structure function is a very basic function since it
can be calculated immediately once we are given the transition probabilities.
Thus the structure function is the function that tells the difference between a random walk on two separate lattices, or between one type of random walk and another type on the same lattice.
For example, in the lOp
case where we have the probability of a step to the left or to the right
equal to
~,
then the structure function would be
=
(10)
cos
Now, once we know the structure function, then we can get the function
u(z,(2TIE/N»,
and thus by Fourier inversion we can get the generating
function
N-l
p (~,z)
=
N-l
r =0
d
Therefore, from this we get
-2TIir's/N
-2TIr
l-zA(-=)
e
L .•.
N
(11)
8
N-l
P t(s)
L ...
ld
=
N
N-l
e
r =0
1
-2nir·s/N
--
(12)
r =0
d
Hence, it follows that for a given random walk, all we need study is its
structure function.
From this structure function, we can get all other
interesting functions and probabilities.
We can extend these notions to the case of the infinite lattice, i.e.,
In this case, we must convert the above sums into integrals.
as
We do so by
l~tting
.
o
1
N
e.
J
Then
l'
2n
2n
N
2nr.
=l ---...J..
=
-.
N
•
de.
J
-2n
=
N
•
and we get
1
P (~, z)
(2n)d
P t(~)
=
1
(2n)d
J.:~. f
J.:~. f
e-ie~ dd e
(13)
1 - z,\(e)
(A(e))t e ies dd e
[Note that the hmits of the integrals could also be
.
(14)
( -n , n) . ]
In the above notation we see that
e2
l+i.!I~(~)--; L~2p(~
when
e
is very small, i.e., when
r
is close to the origin.
(15)
9
Pt(~)
We now want to relate the probabilities
s
on the
t-th
step to the probabilities
first time on step
t.
F (s)
t
of reaching a point
of reaching
-
s
for the
It is easy to see that
t
P (s)
t -
=
0
0
~,Q
t, 0
L
+
(16)
F.(s)P
.(0).
J t-J-
j=l
The generating function for
F (s)
t
is
-
00
(17)
t=l
Now
t
.
t
t -
t~l
t
.
zJ F . (s)p
. (O)z -J
J t-J-
z P (s)
t~l
t~l
j=l
or
P(~,z)
P(~,z)F(~,z)
- 0
~,Q
and hence
P(s,z) - 0
F(~,z)
-
0
~'-
P (Q, z)
1 _
1
P (Q, z)
if
s = 0
(18)
P (~, z)
if
~
f: Q
P(~, z)
Therefore we see that we can get the generating function
know the generating function
F(~,z)
once we
P(~,z).
Further, we can get information about the Po1ya problem from these
first probabilities.
Indeed
10
Prob. that walker returns to origin
F(Q,l)
1 -
{
<
1
P (Q, 1)
(19)
1
if
P(O,l)
00
1
if
P (0,1) <
00
Thus, the walker will be certain to return to the origin if
and the probability of return will be Zess than one if
P(O,l)
Now consider equation (11) which is an expression for
finite lattice.
one.
If
then
r = 0
e
and
P(O,l) =
-2TTiros/N
- -
00
is finite.
P(~,z)
for a
will equal
Therefore,
P (~,z)
where the correction term
Hence if
z=l,
then
¢(~,z)
P(~,z)
1
-d-=-N (1-z)
+
<P(~,z)
is a sum omitting the origin
will diverge, i.e.,
P(Q,l)=oo.
r=O.
Therefore,
for a finite lattice, no matter what dimension, the random walker is certain to return to the origin.
Now we consider the Polya problem for an infinite lattice.
From (13)
we see that
P (Q, 1)
But from (15) it follows that as
_1
d
(2 TT)
:
e -+ 0
ie
f.~o~.f l-;\(e)
then
,
!
I
where
~
The integral can/be expressed as the sum of two
!
11
components, the first being over a
dD
sphere of a small radius,
a,
whose center is at the origin,and the second the integral over the
hypercube of volume
(2n)d
dD
with the central small sphere excluded.
In a
region not including the origin. the integrand is well behaved; no divergence can come from the second component of the integral.
We calculate the
contribution of the integral over a spherical shell about the origin of
e
our
sphere of radius
r,
e=o
space in the neighborhood of
E
and then let
E
~
O.
but omit the contribution of the
As long as the exterior radius,
of the shell is small, the integrand depends only on
polar coordinates in our integration and
dd e
r
so we can use
r
is proportional to
d-l
dr.
Our required integral is then proportional to
E
a
f
I(E)
r
r
E
As
E
~
gration,
P(O,l)
0
a
-2
if
d
1
if
d
2
if
d
a
d-l
f
dr
2
-2
r
d-3
- log E/a
dr
E
a
d-2
d-2
-E
d - 2
~
3
so that the neighborhood of the origin is included in the inteI(E)
~
00
when
diverges when
to Polya's result:
d
d
=
= 1,2
1,2, while
I(E) <
~
00
when
d
3.
Hence
and converges when
d~3.
This is equivalent
A random walker who walks so that steps are only to
nearest neighbor sites is certain to return to his point of origin if he
walks on a lD or 2D lattice.
A nonvanishing escape probability exists in
infinite lattices of more than two dimensions.
Now consider the special case of a 2D square lattice in which steps
can be made to nearest neighbors only, and all possible steps have equal
probability.
below shows,
We wish to calculate the structure function.
As the figure
12
(0,1)
(1,0)
(-1,0)
(0, -1)
there are only four possible step vectors,
~
(0,-1)
s
and
and, if
(0,1).
Thus each step vector
8 = (8 ,8 ),
1 2
A(~)
= (sl,s2) = (1,0),(-1,0),
has probability
p(~)
= ~
then the structure function is
I
p(~)ei~.~
s
(20)
where we let
c.
1.
cos 8.
i
1.
(21)
1, 2, ...
Thus, from (10) and (21) we see that for a simple cubic lattice (S.C.)
of
d=l,2
dimensions we have
A( 8)
for
d
1
for
d
2.
Indeed it can be shown that, in general, for a simple cubic lattice of
dimensions, its structure function is
A(~)
d
13
Further, for a 3D face-centered cubic lattice (F.C.C.), i.e., a cube
with an extra point in the center, the structure function is
Likewise, for a 3D bpdy-centered cubic lattice (B.C.C.), i.e., a cube with
extra points on the faces, we have
Calculating the Polya probabilities for these examples proceeds as
follows.
We know that
probe of return to origin
1
1 - P(O,l) •
The integrals are
for S.C. lattice
P (0,1)
for F.C.C. lattice
for B.C.C. lattice.
Fortunately the values of these integrals have been calculated by G. N.
Watson [9J and hence we have
probe return to origin
~
0.34
S.C. lattice
~
0.28
B.C.C. lattice
~
0.256
F.C.C. lattice.
The number of nearest neighbors to a lattice point in a S.C. lattice is 6;
14
in a B.C.C. lattice, 8; and in a F.C.C., 12.
Thus, it is not surprising
that the return probability diminishes as the number of nearest neighbors
increases, since more ways for escape exist.
3.
FIRST PASSAGE TIMES
We now investigate the average number of steps required to return to
the origin or to first reach a point of the lattice.
We shall first do the
analysis for a finite lattice and then see what happens as we let the number of lattice points become very large.
In order to calculate the first
passage time, we shall need certain moments; specifically, we let
be the average time to first reach point
s.
In terms of
F(~,z)
<n(~»
this
can be written
dF(~,z)
<n(~»
(22)
dZ
Recall that we can write
P (~, z)
where
cP
1
-d--N (l-z)
+
cP (~, z)
has no singularities at
z=l.
(23)
Thus, substituting (23) into (18),
then from (22) we get:
(1)
for
s = Q,
<n (g) >
d
dZ
-
1 -
1
1
+ HQ,z)
d
N (l-z)
z=l
d
d
[1 - N (l-z)]z=l
dZ
d
N
(24)
15
and,
(2)
for
~
I Q
=
d {P(~,Z)}
dZ
p(Q,z) z=l
d
d {l+N (l-Z)¢(~'Z)}
dZ
d
l+N (l-z) ¢(Q, z) z=l
d~
{I + Nd(l-z)[¢(~,z)-¢(Q,z)] + O(1-Z)2} z=l
¢(~,z)] .
(25)
Notice that the expected number of steps required to return to the origin
the total number of lattice points, independently of the structure
is
of the lattice.
As
N becomes ve ry large, it can be shown that
(a)
For ID,
<n(s»
(b)
For 2D,
<n(.Q) >
2
N
(c)
For 3D,
n(s)
3
N
s (N-s)
Ns
log A
1T0 0
l 2
¢ (0,1) -
1
1T0 1 °
2° 3A
where
2
o.
1
=
and
For the derivation of these and other results, reference should be made to
[7].
.
16
4.
NUMBER OF LATTICE POINTS VISITED
We now consider the question of how many distinct lattice points have
been visited after
continues his walk.
n
steps.
This number obviously increases as the walker
Indeed, we can determine this number by using the prop-
P(Q,z).
erties of the generating function
Let
after
n
S
n
be the expected number of distinct lattice points visited
steps and recall that
first reaches point
S
n
s
F (s)
t -
is the probability that the walker
on the t-th step.
1 +
L
Clearly,
[Fl(~)+F2(~)+···+Fn(~)]'
n > 0,
~:fQ
so that
6
F (s)
n-
n
where the summation extends over all lattice points except the origin.
follows that
1,
S
2
Now we can relate the generat}ng function
~
1.
of
6
n
n
6(z)
follows:
6 (z)
F (s)
n-
n
to
P(O,z)
as
It
17
L F(~, z)
- F(Q, z)
s
-
+
1
1
(l-z)P(Q, z)
(26)
where we have used (6).
When
n
is large the asymptotic properties of
~(z)
from the asymptotic properties of
as
z
1
+
S
n
can be derived
by employing the fol-
lowing Tauberian theorem:
Let the sum
for all
n.
If as
=
A(y)
y
La
n
exp(-ny)
converge for
y > 0
a
and let
n
>
0
0,
+
A(y)
¢(x) ~ xO L(x)
where
to
00
as
x
+
00,
is a positive increasing function of
and if
°
~
0
and
L(cx)
~
L(x)
as
x
+
x
00;
which tends
then as
¢en)
r (0+1)
Using the asymptotic properties of
P (Q, z)
P (Q, z)
=
(1_z2)-~
{
P(~,l) + K(l-z)~ +
~(s)
z
+
1,
i.e.,
lD
- - log(l-z)
1
and inserting them in (26), letting
as
P(Q,z)
z
=
2D
(27)
3D
exp(-y),
we find that as
(2y)~
lD
IT
1
(-) /log(-)
y
y
2D
l/yP (Q., 1)
3D.
y
+
0,
18
The Tauberian theorem applies directly to our problem if we choose
ID
er
= ~,
L(x)
2~
2D
er
=
1,
L(x)
TI/logx
3D
er = 1,
L(x)
l/P(Q,l)
Hence we obtain the results for the number of distinct lattice points
visited after
n
steps as
S
n
5.
n-r
00
,
(8n/TI)~
ID
TIn/log n
2D
n/P(Q,l)
3D
(28)
LATTICE WALKS WITH CONTINUOUS TIME VARIABLE
All the above work has been done under the assumption that the random
walker takes steps at regular time intervals.
We now wish to remove this
assumption and consider the case where the time interval between steps has
a continuous probability distribution.
In this theory we are primarily
interested in certain moments of the time variables.
To be specific, assume the steps are made at random times
< •••
0
t
l
<
and that the variables
=
=
which correspond to the length of the time intervals, have a common probability density
~(t).
sities recursively by
Define the sequence
{~(t)}
n
of probability den-
19
o(t) ,
(29)
t
J
1jJ (t)
n
n
1jJ(L)1jJn_l(t-T)dT,
~
1.
o
These are the probability densities of the occurrence of the n-th step at
time
t.
It turns out that the most interesting features in this theory can be
gotten by considering the Laplace transforms of the densities.
densities
1jJ (t)
n
Indeed, the
have as Laplace transform
L1jJ
(Ll)Jn_l) L1jJ
n
(Ll)Jn_2) (L1jJ)
......
=
2
(L1jJ)n
[l)J*(u))n
where
00
1jJ*(u)
f0 e
-ut
1jJ( t) dt.
Now the probabilities which we are interested in here are precisely
F (~, t)
Since
1jJ (t)
n
prob. walker is at
s
at time
t
prob. walker is at
s
for first time at time
t.
is the probability that the n-th step is made at time
and recalling that
first time after
F (s)
n-
n
is the probability that
s
steps, it follows that
F (s) 1jJ (t) .
nn
n~O
Hence its Laplace transform is
t,
is reached for the
20
LF
F(~,u)
F (s) [1jJ*(u)]n
nn~O
F(~,1jJ*(u))
where
F(~,z)
is the generating function for
Ft(~)'
The above formulas can be used to easily derive the moments of various
interesting quantities.
For example, the expected first passage time to
is
t
=
d
- -dU
F(s
,I,*(U))
-,'I'
u=O +
nF (s) d1jJ*(U) [1jJ*(u) ]n-l
n dU
n~O
u=O
+
since
Further the variance of the first passage time to
-2
t
where
[ <n 2( s»
s
-2 + <n(s»
- <n(s» 2 ]T
is
[T 2 - -2]
T
,
T and T2 are the moments of the times between steps.
Most of the formulas and results gotten for the discrete case can be
calculated also for the case of continuous time intervals.
As an example,
for lD we can use a Tauberian theorem for Laplace transforms to find that
the average number of distinct points visited in time
t
is
s
21
[:;t .
S (t)
Other results are obtained in a similar manner.
6.
RANDOM WALKS IN PRESENCE OF DEFECTS
Until now we have been discussing random walks on perfectly periodic
lattices.
We now wish to investigate the effect of a small number of
defective lattice points on the random walk.
For example, we might be
concerned with the effect of "traps" on the probability of the walker's
going from the origin to a given lattice point.
A typical problem of this
sort is the process of photosynthesis, in which there are traps where the
random walker "disappears".
Consider the case in which the lattice contains a few defective points.
As discussed in Section 2, the probability that the walker is at point
after
t
steps is determined by
=
L
p(~,~')
--
only of
s
t-
is the transition probability of a step from
by a walker known to be at
where
(30)
p(s,s')P (s')
s'
where
s
s.
to
s'
However, this probability now depends on
is, so we cannot make the assumption that
(~-~').
s
p(~,~')
is a function
However, we can write
(31)
where the component
p(~-~')
is a correction factor.
a small number of points.
is that of the perfect lattice and
We assume that
q(.,')
q(~-~')
is non-vanishing for only
Since the walker must certainly be at some point
22
after each step, then
I,
for all
s'.
for all
s '.
However, for the perfect lattice we had
and hence
L q(~,~')
o,
s
Define the generating function
G(~,z)
for the system with defects as
t
G(~, z)
(32)
P (s)z .
t
-
As before, we get
G(~,z) -
zL
p(~-~')G(~',z)
6
~,Q
s'
+ z~
~,
q(~,~')G(~',z)
(33)
or equivalently
where
8
is the appropriate operator.
He can use the standard Green's
function technique to solve this, and we get
G(~,z)
L
P(~-~',z)F(~')
s'
P(~,z)
+
zL I
s' s"
P(~-§_' ,z)q(~' ,~")G(~" ,z).
(34)
23
Now suppose that
~
(1)
G(~
,~
(m)
(2)
(m)
""'~
,z),
.
q(',')
is non-vanishing only at the defect points
(1)
G(~
If we can determine
,z), G(~
then we can use these in (34) to find
(2)
G(~,z)
,z), ... ,
in general.
From (34) we get
G(~
(1)
,z)
G(~ (2) ,z)
(1)
+ all
G(~
p(~(2) ,z) + an
G(~
P(s_
,z)
--,-.r
etc.
(1)
(1)
,z)
+ a 12 G(~(2) ,z) +
,z)
+ a 22
G(~
(2)
,z)
+
...
.,
.
'~)
where
zL
a ..
1J
P(~
(")
1
_~' ,z)q(~',~
(")
1
(35)
)
s'
and hence
_P(~(1) ,z)
(1)
(all-l)G(~
,z) + a 12
G(~
(2)
,z)
+ ...
_p (~ (2) , z)
etc.
~
so that if we know the transition probabilities at the defect points, then
we can also find the values
G(~(l) ,z),
G(~(2) ,z),
etc.
As an example, consider the case in which the lattice contains a single
defect point
s
(1)
.
Solving the problem for this case is equivalent to
solving it where the lattice contains an infinite number of defects which
are spaced periodically.
pauses at
s
(1)
Suppose there is probability
E
that a walker
Suppose that on each step the
i.e.,
walker normally goes to one of the nearest neighbors (with equal probabi1ity) and suppose that
s
(1)
has
Z
nearest neighbors.
Then
24
- E/z
if
o
otherwise.
is a nearest neighbor to
s
( 1)
Then,
ZE
')
Z
p (s-s . )G(s (1) ,z)
- -J
-
where the sum in the last term is over all nearest neighbors of
s
( 1)
We
finally get
G(.§.., z)
1 G(.§.. (1) , z)
(1) I
o
.§..,.§..
G(.§..
(1)
j
p(.§..(l) ,z) + E{(z-l)P(Q,z) + 1} G(.§..(l) ,z)
, z)
and hence
P (.§.. (1) , z)
(l-E)+E(l-z)P(Q,z)
d1-z)P(.§..-.§..
P(.§..,z)
Furthermore, as
E ~
G(~(1) ,z)
(1)
(1)
,z)P(~
(37)
,z)
(l-E)+E(l-z)P(Q,z)
1,
~
P (~ (1) , z)
(l-z)P (Q, z)
( 38)
P(.§.._.§..(l) ,z)P(.§.. (1) ,z)
G(~, z)
The case
(=
1
P(~,z)
-
P (Q, z)
corresponds to the situation in which the point
s
( 1)
is a trap.
We proceed to apply the above results to the case in which a 1-dimensional lattice contains one trap at
s
(1)
and we wish to calculate the
25
Folva probability that the walker will eventually return to the origin.
'1f' fore calculating this quantity, observe that we should expect this prob-
ability to increase as the distance,
of
s
(1)
from the origin
increases.
From (19), the probability of eventual return to the origin is
F(O,l)
~ow
from (38) we have
P(O,z) _
G(O,z)
P(_ (1). )P( (1)
s
,z
s
P(O,z)
(P(sO)
I,
,z)I,21
P (0, z)
But bv using (13) it can be shown that
P(s,z)
x
lsi
\..rhere
r---i
[1-\11-z]/z.
x
It
follows that
12 (1) I
G(O,z)
Now as
z
~
1,
then
x
so that
2s(1) i
1
I
I-xl s
I
j
,z
)
26
G(O,l)
and hence the probability of eventual return to the origin in the lD case
is
1 _
F(O,l)
1
12s (1)
Similarly, if we have two traps
s(2) <
a
I
< s(l)
on a lD lattice then
the probability of return to the origin is
F(O,l)
1 -
f,;ls(i) - s(~) I·
Suppose for a lD lattice we have a large number of traps and let the
concentration be
c,
i.e.,
number of lattice points
N.
c
is the number of traps divided by the total
Then it can be shown that the probability of
returning to the origin without being trapped is
1 _
c 1
1
l-c og c
For 2D and 3D this last problem has not been solved.
7.
LATTICE STATISTICS OF DNA MOLECULES
The problem we now consider is one which arises in the chemical analysis
of DNA molecules.
Some of the aspects of this problem are similar to those
of the Ising problem.
A full discussion of this problem can be found in
[5] .
The double-helix structure of a DNA molecule can be thought of simply
as two intertwining strands I'lith hydrogen bonds joining the opposing strands.
27
In fact, there are two such types of bonds joining the two strands, the socalled
A-T
and
G-C
bonds.
Under chemical analysis, such as increases
in temperature and immersion in solvent, these bonds tend to break.
the breaking of these bonds which we wish to study.
It is
Empirical results show
that when most of the bonds of the DNA molecules are of the
A-T
type, then
the strands corne apart at about 60 0 C; whereas, if most of the bonds are
of the
G-C
type, then the strands separate at about 100 0
tical temperature,
T ,
~
we shall mean the point at which
c
C.
By the cri-
of the bonds
in the DNA molecule are broken.
For convenience, the DNA molecule can be drawn as a ladder whose rungs
represent the hydrogen bonds joining the opposite strands.
To make the
analysis easier, we can even think of this chain of bonds as forming a
circle.
For the j-th bond of the ladder define
0.
J
+1
if j-th bond is intact
-1
if j-th bond is broken.
Thus the state of bonding can be characterized by the sequence
(01'02' ... ,0N)
where
N
is the total number of bonds.
We wish to con-
sider probability distributions over these sequences, i.e., distributions
of the form
P(01,02'··· ,ON)
prob. i-th bond is in state
It is obvious that the number of bonds intact is
N
/s( 1+0.)
J
j=l
0i'
i
1, ... ,N.
28
and the number of bonds broken is
N
"
I
~(1-0.
J
)
j=l
It follows that the average number intact is
N
N
<H(l+o
1
.»
J
Av
0=±1 j=l
Also note that if we assume that the bonds break independently then the
distribution function factors:
Now if we define the probability of a bond being intact as being proportiona1 to
exp(-J)
and that of its being broken to
exp J,
we can write
exp(-Jo.)
]
P (0.)
J
2 cosh J
prob. j-th bond is in state
o .•
J
Hence
<I~(l+a. »
J Av
N
I
0=±1
-Jo
e
(1+0)
2
2 cosh J
-J
Ne
2 cosh J
~ [1 - tanh J].
Now if we let
J
1
be the parameter associated with the A-T
bonds, and
J
2
with the G-C bonds, then the number of bonds intact in a system with N A-T
1
bonds and
N
2
G-C bonds is
29
or the fraction of bonds intact is
is the fraction of A-T
where
bonds and
is the fraction of G-C
bonds.
Now consider the case where we have nearest neighbor correlations,
1. e., we have a Harkov situation in which
2
J
1,
o.
In this case, since
1
J
K exp{- 2 (0+0') + Uoo'}
P(O,O')
where
K
the most general form we can construct is
1
K f(o,o')
is the normalizing constant
N
'\
K
)'
L
'-'
o 1 =+1
-
°N=+1
-
exp{-
IT
J
+ UOjO + }
2 (OJ +0 j+ 1)
j l
j=l
Hence
N
d log K
dJ
-1
K
I
I
o 1 =+1
-
o N=+1
-
IT
(0 1+G 2+. • . +G N)
j=l
N <0>
so that
N
< I~(l+o.»
J
j=l
li(N _
Av
--Ll~~
K).
exp{ ... }
30
If we let
F
be the matrix
f(l,l)
f(l,-l)
e
f(-l,l)
f(-l,-l)
e
F
-J+U
e
-U
e
-U
J+U
then
K
where
Al,A
trace
Z are the eigenvalues of the matrix
F.
Thus, as
N~
oo~
where
A
max
max {A l' A } •
Z
Therefore
<
N (1 _
Z
I-l§ (1+0 . ) >
J
Av
3 log A
max)
3J·
Further, we get the fraction of bonds intact to be
f(J)
+-
As the measure of correlation
sign J
U
]
vanishes, then
the random model result discussed before.
f(J)
reduces to
On the other hand, as
we obtain what is called the strong zipper limit.
U
~
00,
In this case, with the
very strong correlation the breaking of a bond forces the same to occur
with the next bond so that the process proceeds like the opening of a
zipper.
31
following references are the principal papers which have been
the material discussed above. For a more complete list of source
on lattice statistics, one should consult the bibliographical
of these references, especially that of Applied Combinatorial
Mathematics~ edited by Beckenbach.
The
used for
material
sections
~ISHER,
in Reports on Progress of
Physics~
1.
M. E.
2.
P. W. KASTELEYN, in Graph Theory in Theoretical
NATO Summer Institute Lectures, 1966.
3.
E. W. MONTROLL,
Mathematics~
4.
5.
1968.
Physics~
Harary, ed.,
"Lattice Statistics," in Applied Corrihinatorial
Beckenbach, ed., Wiley, New York, 1964, pp.96-l43.
, Brandeis Univ. Summer Lecture Notes, 1966.
, "On the Theory of Coiling and Uncoiling of DNA Molecules,"
J. Math. Physics, 1969, pp.57-68.
6.
, "Random Walks on Lattices," Proc. Symp. 'I-n Appl. Math.
Soc.~ 16 (1964), pp.193-220.
Am. Math.
7.
and G. H. WEISS, "Random Walks on Lattices II," J. Math.
Physics 6 (1965), pp.167-l8l.
8.
G. POLYA, "Uber eine Aufgabe der Wahrscheinlichkeitsrechnung
betreffend die Irrfahrt im Strassennetz," Math. Ann. 84 (1921),
pp.149-l60.
9.
G. N. WATSON, "Three Triple Inte~rals," Quart. J. Math. Oxford
Ser. 10 (1939), pp.266-276.