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Blake Stacey (Rev #2)

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Scratchpad

(After Vollmayr-Lee, 2009)

Layer 1 Time

The diffusion propagator is

G D(k,t)=exp(Dk 2t). G_D (k, t) = \exp(-D k^2 t).

To each trivalent vertex, we associate a factor 2λ 0-2\lambda_0, and each initial vertex gets a n 0n_0. Wave-vector (or “momentum”) conservation applies at each vertex. We can read off the self-consistency condition for the tree-level contributions directly from the diagrams:

a tree(t)=n 0+ 0 tdt 1G D(0,tt 1)(2λ 0)a tree(t 1) 2. a_{tree}(t) = n_0 + \int_0^t d t_1 G_D(0, t - t_1) (-2 \lambda_0) a_{tree}(t_1)^2.

The propagator with k=0k = 0 is just 1. Differentiating both sides of the self-consistency equation yields that the time derivative of a treea_{tree} is the integrand evaluated at tt.

da treedt=2λ 0a tree 2 \frac{d a_{tree}}{d t} = -2\lambda_0 a_{tree}^2

This is just a rate equation for a treea_{tree}. With the initial condition a tree(0)=n 0,a_{tree}(0) = n_0, this has the solution

a tree(t)=n 01+2λ 0n 0t. a_{tree}(t) = \frac{n_0}{1 + 2\lambda_0 n_0 t}.

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