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{ \it Physical Review A, 1991}\else \hfil \UN{{\it Page \folio}}\fi}
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\centerline{\bf FINITE SIZE BEHAVIOR OF
RELAXATION TIMES IN}
\centerline{\bf STOCHASTIC MODELS WITH
ACCUMULATING FIXED POINTS}
\vskip 0.4in
\centerline{\bf R. Bidaux{\rm ,}$^a\;$
B.D. Lubachevsky$^b\;$ {\rm and}$\;$ V. Privman$^c$}
\vskip 0.2in
\NI $^a${\sl DPH/SPSRM, Orme des Merisiers, CEN Saclay,
91191 Gif-sur-Yvette, FRANCE}
\NI $^b${\sl AT\&T Bell Laboratories,
600 Mountain Avenue, Murray Hill, NJ 07974, USA}
\NI $^c${\sl Department of Physics, Clarkson University,
Potsdam, NY 13699--5820, USA}
\vskip 0.2in
\NI {\bf PACS:}$\;$ 64.60.Ak, $\,$ 05.90.+m.
\vfill
\centerline{\bf ABSTRACT}
We report numerical and analytical studies of the finite size time scales in a
system with microscopic stochastic dynamics which in the bulk limit corresponds
to an evolution equation with an accumulation point of critical fixed points. Our
results suggest that the finite size behavior is dominated not by fluctuations but by
the bottle-neck statistical weights at near-fixed-point conditions in a finite-size
system.
\vfill
\NP
\NI {\bf 1. Introduction}
\vskip 0.2in
The detailed description of the emergence of
macroscopic ``hydrodynamic'' equations from the
underlying microscopic deterministic or stochastic
dynamics is of importance in many fields. The scope
and emphasis of discussion vary widely, ranging
from conceptual questions of macroscopic irreversibility
to more technical aspects of quantitative and
qualitative characterization of the discreteness
and finite-size effects. The present work falls in the latter
category. Thus, we report numerical Monte Carlo
studies, supplemented by analytical considerations,
of the finite size behavior in a single-variable stochastic
evolution model with an accumulation point of critical
fixed points.
Single-variable stochastic models find applications in
diverse fields ranging from excited electron-hole
statistics in intrinsic semiconductors to models
of chemical reactions, percolation, and evolution of
epidemics.$^{1-7}$ The precise form of the dynamical
rules depends on the phenomenon and model
considered. In more generally oriented studies,
however, one frequently concentrates on dynamical rules
which have some particularly interesting features.
Consider for instance the simplest ``critical point''
hydrodynamic evolution equation
$$ { d \tilde \rho \over d t } = - \tilde \rho ^ 2
\eqno(1.1) $$
\NI Here $\tilde \rho $ is an average density variable,
$0 \leq \tilde \rho (t) \leq 1 $, and $t$ denotes the time.
The large-time behavior implied by (1.1) involves no
exponential-relaxation time scales (which is the generic
case) but is instead power-law, termed {\it critical},
$ \tilde \rho (\hbox{{\it t}-large}) \sim t^{-1}$.
In the discrete, finite-size ``microscopic'' version, the
dynamics is defined by the transition rules for the
variable $n=0,1, \ldots , N$, where $N$ is the total
system size and $n$ is the number of, e.g., wet sites
in percolation or the number of sick individuals in a
group of the total size $N$ for epidemics, etc. The
fluctuating density is then $\rho = n/N$. There have
been very few systematic studies$^{1,5-6,8}$ in the
framework of the modern scaling theories, of finite-size
(e.g., finite population, $N$) effects which are
particularly profound for critical or near critical
stochastic models for which the leading relaxation times
become infinite or very large as $N \to \infty$. Few
results were also published$^{5,9-10}$ for stochastic models
more complicated than single-variable.
The simplest critical-point dynamics (1.1) was
recently studied$^8$ within the scaling description of the
discrete, finite-$N$ effects. In this work we report an
investigation of a more complicated dynamics which
corresponds to the bulk ($N = \infty $) evolution
equation
$$ { d \tilde \rho \over d t } = - \tilde \rho ^ 2
\sin^2 \!\left( { c \over \tilde \rho } \right)
\eqno(1.2) $$
\NI where we will mostly consider the case
$$ c= 1 \eqno(1.3) $$
\NI Thus, $\tilde \rho = 0$ is an accumulation point of
critical fixed points which correspond to the zeros of the
sine function, $\tilde \rho = c/(\pi k)$,\ $k=1,2, \ldots$.
Note that (1.2) integrates to
$$ {\rm cotan} \!\left[ { c \over \tilde \rho (0) } \right]
- {\rm cotan} \!\left[ { c \over \tilde \rho (t) } \right]
=ct \eqno(1.4) $$
The flow diagrams for (1.1) and (1.2)-(1.3) are shown in
Figure 1. The variation of
$\tilde \rho (t) $ with time is always from
right to left as $t$ increases,
with respect to the direction of the abscissa in
Figure 1, for ${d \tilde \rho \over dt } < 0$.
The fixed points are at ${d \tilde \rho \over dt } = 0$, and for (1.2)
infinitely many of them accumulate at the origin. Thus, any initial value
$\tilde \rho (0) $ in the interval $\left[ \big( (k+1) \pi \big)^{-1},
\left( k \pi \right)^{-1} \right)$ evolves in time to the fixed point value
$\tilde \rho (\infty) = \big( (k+1) \pi \big)^{-1}$. The initial values
in the range $[\pi^{-1} , 1]$ evolve to $\pi^{-1}$.
The reason for the particular choice (1.2) is mathematical:
the dynamics described by this evolution equation
extends the simpler dynamics (1.1) to
having an infinite number of fixed points. Thus, (1.2)
is not really favored on physical grounds, and in fact in most
applications the function on the right-hand side of (1.2)
would be rather a polynomial in $\tilde \rho$.
The outline of the next three sections is as follows.
The microscopic transition rules and the associated
notation and definitions are given in Section 2.
Numerical results for the finite-size time scales are
presented in Section 3. Finally, Section 4 is devoted to
some qualitative and quantitative analytical
considerations and to the concluding discussion.
\NP
\NI {\bf 2. Definition of the Microscopic Transition Rates}
\vskip 0.2in
Discrete microscopic time evolution rules usually allow general changes in the
occupation variable $n$ in the range $n=0,1,\ldots ,N$, on time scales of order 1. It
proves convenient, however, to further discretize the time steps$^{1,8}$ in units
of ${1 \over N^2} \ll 1$ and to consider only local changes $n \to n, n \pm 1$. On
the time scales of order 1 such local steps can cumulatively represent rather
general ``bulk'' ($N= \infty$) evolution equations. It turn out that for the discrete
time steps $\Delta t = {1 \over N^2}$ the proper normalization of the transition rates
is
$$ {\rm Rate} (n \to n-1) = N^{-1} A(n,N) \eqno(2.1) $$
$$ {\rm Rate} (n \to n+1) = N^{-1} B(n,N) \eqno(2.2) $$
$$ {\rm Rate} (n \to n) = 1 - N^{-1} \left[ A(n,N)+B(n,N) \right] \eqno(2.3) $$
\NI where $A(n,N)$ and $B(n,N)$ are intensive-in-$N$, i.e., they approach definite
limiting functions of
$$ \rho \equiv {n \over N} \eqno(2.4) $$
\NI as $N \to \infty$. In fact, we take $A= A(\rho)$,
$B= B(\rho)$. Note that with the normalizations chosen, the variation of $n$
for time steps $\Delta t = {1 \over N^2}$ is very slow: the changes ($\pm 1$) in
the $n$ value only occur a fraction of order $1 \over N$ of the attempts. This has
important consequences for numerical implementation of the stochastic dynamics
considered here, as will be described in Section 3.
The transition rules of the model studied in this work are defined as follows:
$$ A = \left[ (1-y) \rho + y \rho^2 \right]
\sin^2 \!\left( { c \over \rho } \right) \eqno(2.5) $$
$$ B = \left[ (1-y) \rho (1- \rho ) \right]
\sin^2 \!\left( { c \over \rho } \right) \eqno(2.6) $$
\NI At $\rho =0$, we take $A=B=0$. The rates depend on a parameter $y \in [0,1]$,
the significance of which
will be discussed shortly. Note that we take $c=1$ for most of this work. The choice
of the value of $c$ will be taken up in Section 4. The important property for now
is that $c=1$ is incommensurate with the period of the sine-squared function, i.e.,
with $\pi$. As a result, the finite-$N$ evolution has no fixed points with $\rho
> 0$. Of course, the point $\rho = 0$ \ ($n=0$) is an absorbing state of the
stochastic dynamics for finite $N$.
Generally, for transition rules (2.1)-(2.3), the $\Omega$-expansion method of
van~Kampen$^1$ can be used to show that the bulk hydrodynamic equation is
$$ { d \tilde \rho \over d t } = B(\tilde \rho ) - A(\tilde \rho ) \eqno(2.7) $$
\NI where $\tilde \rho$ is the bulk average value of the density $\rho$. Note
that the choice (2.5)-(2.6) yields (1.2) for any $y \in [0,1]$.
Usually, the corrections to the average equation (2.7) are due to fluctuations for
finite $N$, and they can be calculated in some regimes by using the
$\Omega$-expansion or other approximation schemes,$^{1,11}$ or by numerical and
scaling-analysis methods.$^8$ Indeed, for the rates leading, e.g., to (1.1),
obtained if the sine-squared factors are dropped in (2.5)-(2.6), it was
found$^8$ that pattern of fluctuations, unlike the bulk average behavior (1.1),
is sensitive to the parameter $y$, both quantitatively and qualitatively.
For a simple stable fixed point at $\tilde \rho = \rho^*$, one can expand
${ d \tilde \rho \over d t } \simeq ( \rho^* - \tilde \rho ) / \tau $, so that the
approach to the fixed point value has a characteristic time scale $\tau > 0$. For
critical fixed points, the leading term in the expansion vanishes ($\tau = \infty$).
One of the most remarkable finite-$N$ effects is that in the microscopic system various
definitions of the time scales characterizing the dynamics near a critical fixed
point yield finite values, usually diverging as powers of $N$, as $N \to \infty$.
For the quadratic fixed point at $\tilde \rho = 0$, with dynamics (1.1), it was
found$^8$ that this divergence, and more generally the form of fluctuation effects,
depend on the value of $y$. Three time scales considered, the ``first passage'' time
to reach $n=0$, the evolution operator spectral-gap time scale (studied$^8$ only for
$y=1$; results for $y=0$ are available for a nearly identical model$^6$), and the time
scale defined by the limits of applicability of the $\Omega$-expansion, --- all
suggested the following pattern. The finite-$N$ time scales varied smoothly with $N$,
and diverged $\sim \!\sqrt{N/(1-y)}$ for fixed $0 \leq y < 1$, as $N \to \infty$,
crossing over to the divergence $\sim \!N$ at $y=1$. The crossover was further described
by a scaling form which will not be discussed here.
The oscillating sine-squared factors added in the rates (2.5)-(2.6) incorporate a
new effect. Indeed, the infinite sequence of the fixed points present in the bulk,
exist for finite $N$ only as approximate fixed points corresponding to small,
``bottle-neck'' transition rates at certain $n$ values. (Recall that we take $c$
incommensurate with $\pi$.) Thus the finite-$N$ time scales can be large both due to
fluctuations near $\rho = 0$ and due to these bottle-neck rates. In fact, the standard
expansion methods$^{1,11}$ are hardly useful here beyond the leading bulk result
(2.7). We will mostly rely on numerical results reported in Section 3; some analytical
considerations will be offered in Section 4. Our conclusion will be that the
``bottle-neck'' effects are dominant.
\NP
\NI {\bf 3. First Passage Times: Numerical Results}
\vskip 0.2in
Perhaps the most profound difference between the finite-$N$ dynamics, defined
by (2.5)-(2.6) with (1.3), and its macroscopic limit (1.2), is that the fixed points
at $\rho = 1/(\pi k)$,\ $k=1,2, \ldots < \infty$ are not exact for finite $N$. As
a result, any finite-$N$ evolution starting from an arbitrary initial value $n(0)$
and advancing according to (2.1)-(2.3), will eventually end up in the only finite-$N$
absorbing state, that at $n=0$. One obvious definition of the ``largest'' time scales
for the finite-$N$ system is therefore via the distribution of the times it takes to
reach $n=0$.
Since the full distribution is difficult to calculate with reasonable accuracy, our
numerical studies were focused on its first and second moments, denoted $T_1$ and
$T_2$, respectively (the precise definitions are given later in this section).
Let us outline the numerical procedure employed.
A {\it time-driven\/} mechanism of simulation would be
advancing time by regular step increments $\Delta t = {1 \over N^2}$.
At each step one would decide
whether the sample process experiences increment
$n \to n+1$, decrement $n \to n-1$ or stays the same $n \to n$.
This easy-to-code time-driven procedure was used, e.g., in Ref. 8.
However, we observed that the transitions rates of {\it changes\/}\ $n \to n \pm 1$
at each given $n$ are quite small, see (2.1)-(2.6).
In the time-driven scheme,
most of the time steps the computer would advance
simulated time without a change to the process.
Thus, we devised a different
{\it event-driven\/} simulation mechanism
wherein the computer efforts are spent
only on the processing of actual events,$^{12}$
and the simulated time is advanced by typically
not equal
time intervals, --- from event to event.
The computations of these, --- potentially much longer than $\Delta t $ time
intervals, --- is done in the following way:
the probability for a change in $n$ to occur exactly at the $k$th
time step ($k=1,2,\ldots $) is $P(1-P)^{k-1}$, where
$$ P=[A(n)+B(n)]/N \eqno(3.1) $$
\NI see (2.1)-(2.2). The weights $P(1-P)^{k-1}$ sum up to 1. Given the current
$n$ value (the initial $n$ value was selected at random) at each step of the
event-driven algorithm, we sample a random number uniformly distributed on (0,1).
It is compared to the partial sums
$$ S_K = \sum_{k=1}^K \left[ P(1-P)^{k-1} \right] = 1-(1-P)^K \eqno(3.2) $$
\NI Depending on which interval $[S_K,S_{K+1})$ the random number was in,
the time was advanced by the appropriate $K \Delta t$. The $n$ value was then
changed by $+1$ or $-1$, with probabilities $B/(A+B)$ and $A/(A+B)$, respectively,
by generating another random number and comparing it to these two weights. The process
was repeated with the new $n$ value until $n=0$ was reached. The comparison of random
numbers to the partial sums (3.2) required a careful consideration of roundoff errors.
The random number generator was the Kirkpatrick-Stoll R250.
The trajectories generated by the new event-driven procedure
are statistically indistinguishable$^{12}$ from those
generated by the time-driven procedure,
while the speed of simulation is at least two
orders of magnitude higher.
Even with the efficient numerical procedure, the simulations required
nontrivial CPU resources. Thus, we had two types of runs. For $N$ values from
1 to 100, we calculated $T_{1,2}$ averaged over 160000 realizations, i.e., different,
randomly selected initial values of $n$, and different random number sequences for the
time evolution, as just described. For these $N$ values, the calculations were
done for $y=0.0,0.1,0.2,\ldots ,1.0$. For $N=101$ to 700, we averaged over only
40000 realizations, and these longer runs were limited to $y=0, {1 \over 2}, 1$.
As a measure of the CPU time, we quote that the long run for $y={1 \over 2}$
took about 3.5 CPU {\it days} on the IBM RS/6000 workstation, model 320.
We denote by $T(i)$ the first passage times obtained in the different runs, where
$i=1,2, \ldots ,I$, with $I=160000 $ or $40000$, as described above.
The moments were defined as follows:
$$ T_1 = I^{-1} \sum_{i=1}^I T(i) \eqno(3.3) $$
$$ T_2^2 = I^{-1} \sum_{i=1}^I T^2(i) \eqno(3.4) $$
Let us first consider the results for $y=0$ which are presented in Figures 2 and 3,
for the first and second moment, respectively. These figures are drawn on a
semilogarithmic scale. The $T_{1,2}$ values do not vary smoothly with $N$. In fact,
they fluctuate widely, sometimes by orders of magnitude, for adjacent $N$ values.
This ``local'' variation is difficult to quantify.
There is, however, also a superimposed overall upward trend with increasing $N$ which
seems to be well represented by $T_{1,2} \sim N$. Note that the data in Figures 2 and
3 were plotted as $T_{1,2}/N$ to emphasize this proportionality. The case $y=0$ for
the simpler model (1.1), without the oscillatory factors in the rates, was studied
extensively,$^{6,8}$ by numerical transfer matrix and Monte Carlo methods, because
it corresponds to the critical point of the mean-field directed percolation. As
mentioned already, the time scales varied smoothly with $N$ and increased $\sim
\!\sqrt{N}$ for large $N$. Obviously, the present, multiple-fixed-point model is
quite different. Here the time scales show behavior which can be represented
as
$$ T \simeq N \Theta (N) \eqno(3.5) $$
\NI where $\Theta (N) $ varies widely and shows no apparent regularities as $N$
increases (at least in the range up to $N=700$ covered by our numerical study).
The second moment, $T_2$, behaves similarly to the first moment, $T_1$, and
is generally somewhat larger. This observation applies for all $y$. Thus, we only
discuss the first moment for other $y$ values. The results of the two other
computer runs up to $N=700$ are shown in Figures 4 and 5, where $y={1 \over 2}$ and
1, respectively. They are qualitatively similar to those for $y=0$. The data follow the
pattern summarized by (3.5). The runs up to $N=100$ for the intermediate $y$ values
(data not shown here) also confirm the conclusion that there is no qualitative
variation of the behavior for $y \in [0,1]$. This is in contrast to the model
(1.1) for which there was a qualitative crossover in the $y$-dependence at $y=1$, as
described earlier.$^8$
The conclusion of our numerical study is that the model with the accumulating
bulk fixed points at $\rho = 0$ is qualitatively different from that with a single
fixed point at $\rho = 0$. Furthermore, for a single-fixed-point model
the finite-size effects were dominated by fluctuations and could be quantified
to a certain extent by using the large-$N$ expansions and scaling ideas.$^{1,8,11}$ For
the new, multiple-fixed-point model the finite-$N$ time scales seem to be dominated by
the bottle-neck weights due to small transition rates when the discretized $\rho$
values are close to the bulk fixed point values. This conclusion is suggested by the
irregularity of the $N$-dependence, and will be further confirmed in the next section.
\NP
\NI {\bf 4. Analytical Considerations and Discussion}
\vskip 0.2in
In an ensemble of systems evolving according to the stochastic dynamics (2.1)-(2.3),
it is convenient to put the probabilities of the various $n$ values at time $t$ in
a column vector ${\cal V}$ of size $N+1$ the entries of which are labeled by
$n=0,1,\ldots ,N$. The appropriate probability vector for $t+\Delta t = t + {1 \over
N^2}$ is then obtained by applying (left-multiplying) the matrix of the transition
rates (2.1)-(2.3) on the vector at $t$:
$$ {\cal V}_{t+\Delta t} = {\cal M} {\cal V}_t \eqno(4.1) $$
\NI The transfer matrix, or evolution operator, ${\cal M}$, has elements
$$ {\cal M} = \pmatrix{\vphantom{\vdots}1&{\rm R}(1 \to 0)&0&\ldots&\cr
\vphantom{\vdots}0&{\rm R}(1 \to 1)&{\rm R}(2 \to 1)&0&\ldots\cr
\vphantom{\vdots}0&{\rm R}(1 \to 2)&{\rm R}(2 \to 2)&{\rm R}(3 \to 2)&0&\ldots\cr
\vdots&0&{\rm R}(2 \to 3)&{\rm R}(3 \to 3)&{\rm R}(4 \to 3)&0\cr
&\vdots&0&\ddots\hfill&\ddots\hfill&\ddots\hfill\cr
}_{(N+1)\times (N+1)} \eqno(4.2)$$
\hphantom{A}
\NI where R abbreviates for ``Rate;'' see (2.1)-(2.3). This tridiagonal, stochastic
matrix has the largest eigenvalue $\lambda_0 =1$, corresponding to the ``stationary''
evolution in the absorbing state $n=0$. All other eigenvalues are less than 1 in
absolute value; some may form complex-conjugate pairs although typically a large
number of the leading eigenvalues are real. The time scales $\xi_m$ associated with
the spectral gaps are obtained by the identification
$$ \left( {\lambda_m \over \lambda_0} \right)^{t/\Delta t} = {\rm e}^{-t/\xi_m}
\eqno(4.3) $$
\NI where we assumed for simplicity that $\lambda_{m>0}$ is real. For the present
case, we have
$$ \xi_m = - {1 \over N^2 \ln \lambda_m } \eqno(4.4) $$
For the mean-field directed percolation model,$^6$ which is nearly identical
with the model (1.1) at $y=0$, as formulated here, with no sine-squared factors
in the transition rates, the time scales $\xi_m$ had a physical interpretation of the
longitudinal percolation correlation lengths, and the largest such length was studied
numerically quite extensively.$^6$ With oscillating rates, the numerical methods
employed$^6$ become unstable to roundoff errors, and no results for large $N$ values
can be obtained. Analytical progress can be made, however, for the special case $y=1$.
Indeed, for $y=1$ the transition rate for $n \to n+1$ vanishes; see (2.6). Thus the
lower off-central diagonal in (4.2) consists of zeros. As a result, ${\cal M}$ is an
upper-diagonal matrix, and the eigenvalues are given by the diagonal elements:
$$ \lambda_m = 1-{m^2 \over N^3} \sin^2 \!\left({cN\over m} \right)
\eqno(4.5) $$
\NI where $m=0,1,2,\ldots ,N$, and we used (2.3)-(2.5).
An interesting question arises: which of the eigenvalues (with $m>0$) is the largest
for a given $N$, yielding the largest time scale $\xi$? More generally, one could
consider the distribution of the $\xi$ values for each $N$. We haven't attempted
the mathematical analysis of this rather complicated problem in the number theory.
Indeed, it is obvious that the answer depends on how close $c$ is to a number
commensurate with $\pi$. By a numerical test up to $N=10^4$ we found that for our
preferred choice $c=1$ the largest $\xi$ value is obtained for $m=1$. However, for
other $c$ values this need not be the case. As indicated, the proximity to a number
commensurate with $\pi$ seems to be decisive. For instance, for $c=1.570800$ the
largest eigenvalue is obtained with $m=1$, for $N=2, 3, \ldots, 100$, as one can
easily check numerically. However, for $c=1.570796$, which is closer to $\pi /2$,
the largest eigenvalue up to $N=100$ is obtained for $m=m_{max}>1$ for several
$N$ values: 6, 10, 12, 20, 24,
30, 36, 40, 42, 48, 50, 54, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100. The corresponding
$m_{max}$ values are 3, 5, 3, 5, 3, 15, 9, 5, 3, 3, 25, 9, 15, 11, 35,
9, 13, 5, 3, 3, 3, 25.
As mentioned, we haven't attempted a mathematical explanation of the patterns
observed. A conclusion for the case $c=1$ is that at least up to $N=10^4$,
the largest time scale is given by the analytical expression
$$ \xi_1 (N) = - \left[ N^2 \ln \left( 1- N^{-3} \sin^2 N \right) \right]^{-1}
\eqno(4.6) $$
\NI Note that for $N \gg 1$, one can use
$$ {\xi_1 (N) \over N} \simeq {1 \over \sin^2 N } \eqno(4.7) $$
\NI The result (4.6) is plotted for $N=1$ to 700 in Figure 6, and it should be compared
with the first-passage time shown in Figure 5. The physical interpretation of the
transfer matrix time scales $\xi_m$ for $y=1$ is less interesting than that of the
first-passage times considered in Section 3. Indeed, while the first-passage times
provide an overall characteristic time of the stochastic dynamics, the largest
transfer matrix times in this case ($y=1$) simply translate to the time scales the
``bottle-neck'' transition rates at densities $\rho = {m \over N}$. The
largest such time scale is given by (4.6) and is obtained at the lowest possible
discrete $\rho$ value, $1 \over N$ (for $c=1$).
However, as far as general features
of the increase of the time scales as $N \to \infty$ go, the transfer matrix results
do yield a useful insight. Indeed, the overall trend $\sim
\!N$ is seen, and one can write a representation (3.5) for $\xi_1 (N)$. The
corresponding varying function $\Theta (N) \simeq \left( \sin N \right)^{-2}$, which is
actually plotted in Figure 6, on a semilogarithmic plot, shows an interesting pattern
which is not exact and probably reflects the incommensurability of $c=1$ with $\pi$.
The pattern is self-similar in that it is observed ``rescaled'' in plots for much
larger $N$ values. It is not clear to what extent this visually observed semi-regular
behavior can be utilized to quantify the results for the first-passage times, e.g.,
Figure 5, which look much more irregular.
In summary, the main conclusions of our work are that for accumulating fixed
points, the finite-$N$ time scales are dominated by the bottle-neck near-fixed-point
transition rates. Large fluctuations in the time scale values result, for
adjacent $N$, and they are difficult to quantify in general. The overall trend $\sim
\!N$ in the finite-$N$ time scales was found numerically and confirmed analytically
for a special case. A weak dependence on the values of the parameter $y$ suggests that
the r\^ole of fluctuations, which are important at isolated fixed points, is secondary
in our case.
\NP
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\vskip 0.2in
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\NP
\centerline{\bf FIGURE CAPTIONS}
\NI\hang {\bf Figure 1.} The ``flow diagram"
corresponding to (1.2)-(1.3), solid line, and to
(1.1), broken line.
\NI\hang {\bf Figure 2.} The average (first-moment) passage time, $T_1$, to
$n=0$. These numerical data were obtained for $y=0$; see Section 3.
\NI\hang {\bf Figure 3.} The second-moment-average passage time, $T_2$, to
$n=0$. These numerical data were obtained for $y=0$, in the same computer run that
yielded the data shown in Figure 2.
\NI\hang {\bf Figure 4.} The average (first-moment) passage time, $T_1$, to
$n=0$. These numerical data were obtained for $y={1 \over 2}$.
\NI\hang {\bf Figure 5.} The average (first-moment) passage time, $T_1$, to
$n=0$. These numerical data were obtained for $y=1$.
\NI\hang {\bf Figure 6.} The largest transfer matrix time scale for $y=1$; see
(4.6). Note that the value for $N=355$, $\xi_1 (355)/355 \simeq 10^{9.04}$ is
outside the ordinate limits (chosen to facilitate comparison with Figure 5).
\bye