Statistical Field Theory and Applications

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Statistical Field Theory and Applications M2 - Theoretical Physics ENS-Paris, 2019-2020 Denis BERNARD: [email protected] Jesper JACOBSEN: [email protected] What are we aiming at describing? * Systems with infinitely many degrees of freedom (d.o.f.) : —> Collective phenomena, collective modes, geometrical (random) patterns,…. * Typical (stat. mech.) examples : Percolation: the simplest local model Ising spin clusters: - black = filled - white = empty - green = spin up - red = spin down. * More (geometrical) examples: Self-avoiding walks, alias « polymers » Brownian motion (2D). Critical phenomena, universality,… Renormalisation group,… Aspects of random geometry,… How are we describing them? * Statistical Field Theory : — Part of statistical physics (at equilibrium or not)… with ramifications in other domains (say Condensed Matter, QFT,…) — An important tool of theoretical physics with deep math connections,… (probability theory, representation theory, geometry, RG,…) Samples of a (discretized) Gaussian free field Statistical physics with infinitely many (geometrical… singular) degrees of freedom. Plan : (order different from the lecture notes) — Brownian motions, random paths. criticality & — Statistical lattice models and field theory. } universality — Renormalization group and universality. — Statistical field theory: free theory. field theory : — Statistical field theory: interactions. } (objects, tools & techniques) — Conformal field theory: basics. — Scaling limits, field theory scaling limits & RG and the renormalization group. } — 12+1 sessions: 3h-lecture + 1h-exercises. Exercises (1 week after the lectures). — Lecture notes : see https://www.phys.ens.fr/~dbernard/ « Statistical field theory and applications: an introduction for (and by) amateurs » — Exercice booklet (with corrections): (address=idem).

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