4.3. Other random walks[[Student version, December 8, 2002]] 113
0.30.40.50.60.81
126000 10^43. 104 5. 104Rg, μmN, basepairsyccdLight0
01cmxcm,μma bd
Figure 4.8:(Schematic; experimental data; photomicrograph.) Experimental test of the self-avoiding random-walk
model of polymer conformation, in two dimensions. (a)Experimental setup. A negatively charged DNA molecule
sticks to a positively charged surface. The DNA has been labeled with a fluorescent dye to make it visible in a
light microscope. (b)The entire molecule performs a random walk in time. The plot shows its center of mass on
successive observations (compare Figure 4.2b,c on page 102).(c)Successive snapshots of the molecule taken at 2s
intervals. Each one shows a different random conformation. The fine structure of the conformation is not visible, due
to the limited resolving power of an optical microscope, but the mean-square distance of the molecule from its center
of mass can still be calculated. (d)Log-log plot of the size of a random coil of lengthNbasepairs versusN.Foreach
N,the coil size has been averaged over 30 independent snapshots like the ones in (c) (see Figure 4.5). The average
size increases proportionally toN^0.^79 ±^0.^04 ,close to the theoretically predictedN^3 /^4 behavior (see Problem 7.9).
[Digital image kindly supplied by B. Maier; see also (Maier & R ̈adler, 1999).]
plete knowledge. Each bases his decisions on the aggregate of the other investors’ decisions, as
well as the totally unpredictable events in the daily news. How could we possibly say anything
predictive about such a tremendously complex system?
Indeed, we cannot predict an individual investor’s behavior. But remarkably, the very fact that
investors are so well informed about each others’ aggregate behaviordoeslead to a certain statistical
regularity in their behavior: It turns out that over the long term,stock prices execute a random
walk with drift.The “thermal motion” driving this walk includes the whims of individual investors,
along with the natural disasters, collapses of large firms, and other unpredictable news items. The
overall drift in the walk comes from the fact that in the long run, investing money in firms does
make a profit.
Why is the walk random? Suppose that a technical analyst finds that there was a reliable
year-end rally, that is, every year stock prices rise in late December, then fall in early January.
The problem is that once such a regularity becomes known to market participants, many people
will naturally choose to sell during this period, driving prices down and eliminating the effect in
the future. More generally, the past history of stock-price movements, which is public information,