Exponential Decay for Subcritical Contact and Percolation Processes

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“…As in the continuous-time case, we use upper case letters for points in We write P λ ε and E λ ε for the associated probability measure and the expectation operator. It can be proved as in [3] that P λ ε converges weakly to P λ as ε ↓ 0. Therefore the contact process is wellapproximated by the above discretized model.…”

“…(2.1) for some positive constants C i , C i , i = 1, 2, 3 (see [1,2,3] and references therein). The critical infection rate can also be defined as λ c = sup λ : Ξ λ (t) → 0 exponentially as t ↑ ∞ , (2.2) where Ξ λ (t) = X:τ (X)=t φ λ (O, X) (see [11]); when λ > λ c , Ξ λ (t) does not vanish because…”

“…constructing an oriented bond percolation model on Z d ×ε Z which converges to the contact process in a limit ε ↓ 0 (Section 3.1), 2. applying the lace expansion to the connectivity function for this oriented percolation (Section 3.3), 3. estimating the irreducible two-point functions appearing in the lace expansion to obtain a ε-uniform infrared bound (Section 5).…”

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“…For the nearest-neighbor CP and OP and for the spread-out CP and OP with L ≥ 1, it has been proved that β ≤ 1 if the exponent β exists [1,4,6], and that β exists and equals 1 under the triangle condition [3,4]. It has also been proved [13,17] that the exponent ν exists and equals 1/2, and the Gaussian scaling limit …”

“…For the nearest-neighbor CP and OP with d ≥ 1 and for the spread-out CP and OP with d ≥ 1 and L ≥ 1, it has been proved [1,2,4,6] that χ(λ) is finite when λ < λ c and diverges as λ ↑ λ c . In fact…”