Monte Carlo Methods in Statistical Mechanics: Foundations and ...

also of practical importance. b C(t) isanunbiased estimator of C(t), and b C(t) is almost
unbiased (the bias is of order 1=n) [15, p. 463]. Their variances and covariances are [15,
pp. 464{471] [14, pp. 324{328]
1X
var( b C(t)) = 1
n m=;1
+ o 1
n
cov( b C(t) b C(u)) = 1
n
1X
h C(m) 2 + C(m + t)C(m ; t)+ (t m m + t) i
[C(m)C(m + u ; t)+C(m + u)C(m ; t)
m=;1
+ (t m m + u)] + o 1
n
where t u 0and is the connected 4-point autocorrelation function
(3.10)
(3.11)
(r st) h(fi ; )(fi+r ; )(fi+s ; )(fi+t ; )i
; C(r)C(t ; s) ; C(s)C(t ; r) ; C(t)C(s ; r) : (3.12)
To leading order in 1=n, the behavior of b C is identical to that of b C.
The \natural" estimator of (t) is
if the mean is known, and
b(t) b C(t)= b C(0) (3.13)
b(t)
bC(t)= b C(0) (3.14)
if the mean is unknown. The variances and covariances of b(t) and b(t) canbe
computed (for large n) from (3.11) we omitthe detailed formulae.
The \natural" estimator of int would seem to be
b int
? 1
2
n X;
1
t = ;(n ; 1)
b(t) (3.15)
(or the analogous thing with b), but this is wrong! The estimator de ned in (3.15) has a
variance that doesnotgoto zero as the sample size n goes to in nity [14, pp. 420{431],
so it is clearly a very bad estimator of int. Roughly speaking, this is because the sample
autocorrelations b(t) for jtj contain much \noise" but little \signal" and there are
so many ofthem (order n) that the noise adds up to atotal variance of order 1. (For a
more detailed discussion, see [14, pp. 432{437].) The solution is to cuto the sumin
(3.15) using a \window" (t) whichis 1forjtj < but 0 for jtj :
b int
1
2
n X;
1
t = ;(n ; 1)
15
(t) b(t) : (3.16)