# The Azimuth Project Blake Stacey (Rev #2, changes)

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Blake Stacey is a scientist. His blog:

(After Vollmayr-Lee, 2009)

The diffusion propagator is

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

To each trivalent vertex, we associate a factor $-2\lambda_0$, and each initial vertex gets a $n_0$. 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:

$a_{tree}(t) = n_0 + \int_0^t d t_1 G_D(0, t - t_1) (-2 \lambda_0) a_{tree}(t_1)^2.$

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

$\frac{d a_{tree}}{d t} = -2\lambda_0 a_{tree}^2$

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

$a_{tree}(t) = \frac{n_0}{1 + 2\lambda_0 n_0 t}.$

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