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2. To describe the growth quantitatively, we introduce two functions: • hi (t) denotes the height of the interface at time t and lattice site i. 14) i=1 characterizes the ”roughness“ of the surface. Numerical experiments show that w is a function of the systemsize L and time t and scales w(L, t) ∝ tβ t Lα t tx tx with tx ∝ Lz . 3). This function can be fitted. g. at t ≈ tx , we have w ∝ tβx ∝ Lzβ and w ∝ Lα what implies α = zβ. The existence of power laws and a scaling function strongly reminds us of RG!

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.

We will now define the reduced free energy (per spin): f = − N1 ln Z. Then Z = e−N f ({k}) The partition function Z of a system is invariant under RG Z = Tr {s} e −H[s] = Tr {s } e −H [s ] =Z Thus, Z = const × e−N f ({k}) with N = b−d N with f ({k}) = g({k}) + b−d f ({k }) g({k}) represent the ”analytic first modes“ and f ({k}) = b−d f ({k }) defines the singular behavior around fixed point. For the universality class of the Ising model ut , uh are relevant (yt , yh > 0) and f (ut , uh ) = b−d f ( ut , uh ) = ...

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### Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12) by Johannes Berg, Gerold Busch

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