Isoperimetric lower bounds for critical exponents for long-range percolation

We study independent long-range percolation on Z^d where the vertices x and y are connected with probability 1 − e⁻β⃦x−^{y⃦−d−α} for α > 0. Provided the critical exponents δ and 2 − η defined by δ = lim_{n→∞ log}(_P⁻_βc^log₍|_K⁽ⁿ⁾_0|≥n)) and 2 − η = lim_x→∞^log(^{Pβc (0↔x))} + d exist, where K₀ is the cluster containing the origin, we show that log(⃦x⃦) _δ≥ d + (α ∧ 1) and 2 − η ≥ α ∧ 1. d − (α ∧ 1) The lower bound on δ is believed to be sharp for d = 1, α ∈ [ ¹₃ , 1) and for d = 2, α ∈ [ ²₃ , 1], whereas the lower bound on 2 − η is sharp for d = 1, α ∈ (0, 1), and for α ∈ (0, 1] for d > 1, and is not believed to be sharp otherwise. Our main tool is a connection between the critical exponents and the isoperimetry of cubes inside Z^d.