By Johannes Berg, Gerold Busch

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The occupied bonds form a graph on the lattice. Summing over {σi } only terms represented by closed graphs contribute to Z. 9) h=0 N N N Compare this with the result obtained above: Z = λN 1 + λ2 = (2 cosh k) + (2 sinh k) . Correlations can also be calculated σm σn = 1 Z σm σn σi (1 + σm σn t) * only graphs with an even number of occupied bonds at all nodes except m and n contribute. m and n must have an odd number of occupied bonds. 7 The Ising model in 2 dimensions This section closely follows the exposition in Kardar, which is based on an approach pioneered by M. *

Divide a time interval T in many small intervals ∆t. The probability not to jump away in a time interval ∆t is (1 − a∆t), the probability not to jump away in the intervall T is then (1 − a∆t)T /∆t → e−aT . This means, the probability to make a jump at some time is = 1. e. the probability for the final position x to be finitely different from z goes to zero faster than ∆t goes to zero. 1) |x−z|< which is called a ”drift“. 2) |x−z|< which is called ”diffusion“. g. 1 ∆t dx(x − z)3 p(x, t + ∆t|z, t) ≤ |x−z|< < 1 ∆t ∆t ∆t→0 → dx|x − z|(x − z)2 p(x, t + ∆t|z, t) |x−z|< dx(x − z)2 p(x, t + ∆t|z, t) |x−z|< B(z, t) = O( ) Hence, knowing drift and diffusion are sufficient to characterize the system.

1 t-t0 = 1. t-t0 = 10. 1: Wiener process: The figure shows the spreading of an initially sharp peaked distribution. 2 Stochastic differential equations For the master equation, we specified w(x, x ). The equation depends on the full w(x, x ) but we knew how to generate sample paths. For continuous processes, the first two moments of the propagator were sufficient. But how to generate trajectories? 9) where ξ(t) is a random (”stochastic“) function, a(x(t)) denotes the drift, b(x(t)) the diffusion.