The two regimes

Lemma 10.3 · lem:tworegime · page 27

What is happening

The cutoff dynamics has an exact light cone: by Proposition 2.7, β^g_t(A) lies in A(O_|t|) whenever A ∈ A(O). So as |t| grows, the observable spreads at unit speed and nothing outside the cone has heard of it yet.

Take a cell B_j at distance r_j from O. While |t| ≤ r_j/2 the cone has not reached it — in fact dist(B_j, O_|t|) ≥ r_j − |t| ≥ r_j/2 — and the clustering lemma applies with the uniform gap γ, giving a bound that is exponentially small in r_j. Past that time the argument is gone and only the trivial bound ‖V_jΩ‖·‖A‖ ≤ 2M is available.

Drag t across r_j/2 and watch which bound is in force. The whole content of Lemma 10.3 is that both are available simultaneously, so the one-path estimate can integrate the good bound over the short times and pay the trivial one only over the rest. Summing that trade-off over the cells is what produces the rate function Ψ_γ of Lemma 10.5.

Statement and proof: result inventory. The constant M is the uniform local moment of Theorem 5.1; γ is the uniform gap of Theorem 7.8.