Manuscript Oai212-A@1 · Oai212-A.tex · section · 79323841

Classical first-passage process and finite-volume control

Classical first-passage process and finite-volume control · lines 199–218

Let

E_{2} = {{u, v} : u, v \in Z^{2}, ∣ u - v ∣_{1} = 1} .

Attach independent weights $τ_{e} \sim Exp (λ)$ to $e \in E_{2}$ . For a finite nearest-neighbour path $γ$ , define

τ (γ) = e \in γ \sum τ_{e}, T (u, v) = γ : u \to v in f τ (γ) .

Since weights are positive, loops can be erased without increasing cost. Let

I (t) = {x : T (0, x) \leq t}, A (t) = x \in I (t) ⋃ (x + [- \frac{1}{2}, \frac{1}{2}]^{2}) .

Displayed formulas in this passage

unlabeled · line 202
$E_{2} = {{u, v} : u, v \in Z^{2}, ∣ u - v ∣_{1} = 1} .$
unlabeled · line 207
$τ (γ) = e \in γ \sum τ_{e}, T (u, v) = γ : u \to v in f τ (γ) .$
unlabeled · line 213
$I (t) = {x : T (0, x) \leq t}, A (t) = x \in I (t) ⋃ (x + [- \frac{1}{2}, \frac{1}{2}]^{2}) .$