PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

170 Geoffrey Grimmett
Theorem 5.5 (Holley’s Inequality). Let µ1 and µ2 be positive probability
measures on (ΩE, FE) such that
(5.6) µ1(ω1 ∨ ω2)µ2(ω1 ∧ ω2) ≥ µ1(ω1)µ2(ω2) for all ω1, ω2 ∈ ΩE.
Then
µ1(f) ≥ µ2(f) for all increasing f : ΩE → R,
which is to say that µ1 ≥ µ2.
Proof of Theorem 5.5. The theorem is ‘merely’ a numerical inequality involving
a finite number of positive reals. It may be proved in a totally elementary
manner, using essentially no general mechanism. Nevertheless, in a more
useful (and remarkable) proof we construct Markov chains and appeal to the
ergodic theorem. This requires a mechanism, but the method is beautiful,
and in addition yields a structure which finds applications elsewhere.
The main step is the proof that µ1 and µ2 can be ‘coupled’ in such a
way that the component with marginal measure µ1 lies above (in the sense
of sample realisations) that with marginal measure µ2. This is achieved by
constructing a certain Markov chain with the coupled measure as unique
invariant measure.
Here is a preliminary calculation. Let µ be a positive probability measure
on (ΩE, FE). We may construct a time-reversible Markov chain with state
space ΩE and unique invariant measure µ, in the following way. We do this by
choosing a suitable generator (or ‘Q-matrix’) satisfying the detailed balance
equations. The dynamics of the chain involve the ‘switching on or off’ of
components of the current state. For ω ∈ ΩE, let ωe and ωe be given as in
(4.1). Define the function G : Ω2 E → R by
(5.7) G(ωe, ω e ) = 1, G(ω e , ωe) = µ(ωe)
µ(ω e ) ,
for all ω ∈ ΩE, e ∈ E; define G(ω, ω ′ ) = 0 for all other pairs ω, ω ′ with
ω �= ω ′ . The diagonal elements are chosen so that
It is elementary that
�
G(ω, ω ′ ) = 0 for all ω ∈ ΩE.
ω ′
µ(ω)G(ω, ω ′ ) = µ(ω ′ )G(ω ′ , ω) for all ω, ω ′ ∈ ΩE,
and therefore G generates a time-reversible Markov chain on the state space
ΩE. This chain is irreducible (using (5.7)), and therefore has a unique invariant
measure µ (see [169], p. 208).