value of .s (x) f(x), the < value of an operator f defined on the state-space,
under some distribution t. 'I : full sum i: not be feasible to : .orrn when S is large
(as we shall see, this is most certainly true in the u work!). T. idea is, then,
to construct a. Markov chain which has as its equilibrium distribution and perform
ortance : pling to obtain an approximation of the expectation value. So in Monte
Carlo simulations we are given a distribution Fr and it is the Markov chain we want to
find. Ti : stands in contrast to .. other .. plications of the Markov theory, where
the transition matrix is known, and the equilibrium distribution is the object of interest.
Of course we will not be able to find the Markov chain in itself; instead we find what we
shall call a Markov sequence: a sequence of actual states, rather than random variables in
the state-space. T : is : table because if the sequence really represents the Markov
chain, then the states will distribute as dictated '. the random variables T.
central trick in a Carlo simulation is how to obtain the Markov sequence from
the distribution rr. Although sometimes very difficult in practice, the idea is almost
embarr;. '.. simple. Since Fr satisfies the Ergodic .....em, rr(x,) 74 0 for all x .
P 1 the most straight forward ansatz would be to select states x G S at random
and -,ending each to the Markov sequence with probability yr( r). I.. result would
certainly be a. Markov sequence with equilibrium distribution t. T : method, however,
would be hopeless in i- i- cases because of the computational complexity of calculating
7r(x)), for an arbitr x, as many times as would be :' 1 to obtain an acceptably
long Markov .... Fortunately, many important cases in pl exhibit what we
shall call Monte Carlo ..'. or me-' '" for short. T: is the property that the
ratio T(x')/(,T(x) is drastically simpler to evaluate ..-. :: '.onally if x is close to x' in
a sense that is simulation specific. It suffices to that they are close precisely when
x x' and x (x')/r(x) are .:. -. ::i !:. nally simple operations. We :: lively
write x' = x I+ and talk about Ax being small, even though there -- not be an
addition-operation, metric or measure defined on state i. For an minc-local m we