These notes provide a concise introduction to important applications of the
renormalization group (RG) in statistical physics. After reviewing the scaling
approach and Ginzburg-Landau theory for critical phenomena, Wilson's momentum
shell RG method is presented, and the critical exponents for the scalar Phi^4
model are determined to first order in an eps expansion about d_c = 4.
Subsequently, the technically more versatile field-theoretic formulation of the
perturbational RG for static critical phenomena is described. It is explained
how the emergence of scale invariance connects UV divergences to IR
singularities, and the RG equation is employed to compute the critical
exponents for the O(n)-symmetric Landau-Ginzburg-Wilson theory. The second part
is devoted to field theory representations of non-linear stochastic dynamical
systems, and the application of RG tools to critical dynamics. Dynamic critical
phenomena in systems near equilibrium are efficiently captured through Langevin
equations, and their mapping onto the Janssen-De Dominicis response functional,
exemplified by the purely relaxational models with non-conserved (model A) /
conserved order parameter (model B). The Langevin description and scaling
exponents for isotropic ferromagnets (model J) and for driven diffusive
non-equilibrium systems are also discussed. Finally, an outlook is presented to
scale-invariant phenomena and non-equilibrium phase transitions in interacting
particle systems. It is shown how the stochastic master equation associated
with chemical reactions or population dynamics models can be mapped onto
imaginary-time, non-Hermitian `quantum' mechanics. In the continuum limit, this
Doi-Peliti Hamiltonian is represented through a coherent-state path integral,
which allows an RG analysis of diffusion-limited annihilation processes and
phase transitions from active to inactive, absorbing states.Comment: 28 pages; 49th Schladming Theoretical Physics Winter School lecture
notes; to appear in Nucl. Phys. B Proc. Suppl. (2012
