exponential decay below the critical point
/ Menshikov: men-SHEE-kov; Aizenman: EYE-zen-man /
What does the subcritical phase actually look like below p_c? Intuitively, with too few open edges, clusters should be small — but small in what quantitative sense? The answer is the sharpest possible: for every p < p_c, the probability that the origin connects to distance n decays EXPONENTIALLY in n, and the cluster size has exponential tails. There is no slow, polynomial 'almost-critical' decay just below p_c; the off-critical behaviour is robustly exponential right up to the threshold. This is the theorem that makes the phase transition sharp.
Precisely: for p < p_c there is a constant psi(p) > 0 (an inverse correlation length) such that P_p(0 connects to the boundary of the box of radius n) <= e^(-psi(p) n) for all large n, and consequently E_p[|C|] < infinity with P_p(|C| >= k) decaying exponentially in k. This was proved independently by Menshikov (1986) and by Aizenman and Barsky (1987), the latter via clever differential inequalities for the magnetization-like quantities. The deep content is that the threshold for 'theta(p) > 0' coincides with the threshold for 'exponential decay of connectivity' — a priori one might fear a sub-phase where clusters are finite but only polynomially controlled; sharpness rules this out. A strikingly short modern proof was given by Duminil-Copin and Tassion (2016) using a clever functional of edge boundaries and Russo's formula, sidestepping the heavier machinery.
Exponential decay is the foundation of the whole subcritical theory: it gives a finite mean cluster size, a well-defined correlation length xi(p) = 1/psi(p) that diverges as p rises to p_c, and the renewal/Ornstein-Zernike refinements that pin down the precise pre-exponential corrections. It is also the input that lets one define critical exponents (xi(p) ~ (p_c - p)^(-nu)). One honest caveat: the theorem gives the exponential RATE but the sharp constant psi(p) and the polynomial prefactor are far more delicate (Ornstein-Zernike theory). And at p = p_c itself the decay is NOT exponential — connectivity decays only polynomially, which is the entire reason criticality is interesting.
At p slightly below p_c, the correlation length xi(p) measures the typical cluster diameter: P_p(0 connects to distance n) is roughly e^(-n/xi(p)). As p increases to p_c, xi(p) blows up like (p_c - p)^(-nu) (nu = 4/3 in 2D), so just below criticality clusters are large but still almost-surely finite with exponential tails.
Below p_c connectivity decays as e^(-n/xi); the correlation length xi diverges at the critical point.
Exponential decay holds strictly below p_c and pins down the exponential rate, but the sharp constant and prefactor need Ornstein-Zernike theory. AT p_c the decay is only polynomial — confusing the two erases the whole point of criticality.