A tantárgy az alábbi témakörök ismeretére épít:
basics of probability theory
A tantárgy szerepe a képzés céljának megvalósításában:
TTK fizikus és matematikus képzések szabadon választható tárgya
A tantárgy részletes tematikája magyarul és angolul:
Percolation theory: definitions and their equivalence. Examples using the Peierls contour method, first and second moment method. The Harris-FKG correlation inequality.
The Ising model on finite graphs: definition, spatial Markov property, basic properties of the partition function, definition of long range order.
Glauber dynamics and other Markov chains. Holley's proof of the FKG-inequality. Infinite volume Gibbs measures.
The FK random cluster model, Edwards-Sokal coupling, the Potts models, Uniform Spanning Tree.
Mean field models: Erdős-Rényi random graph and the Curie-Weiss phase transition.
Pólya’s theorem on recurrence versus transience of simple random walk on Z^d. Green's function.
The Discrete Gaussian Free Field (DGFF) on graphs. Relation to other models, such as Ising.
Intuitive glances at more advanced topics:
Critical point for planar percolation: the Harris-Kesten theorem (1980).
Scaling limits: Brownian motion. Continuum GFF. The conformal invariance of critical planar percolation and other FK models. (Fields medals to W. Werner 2006 and S. Smirnov 2010.)
Mermin-Wagner and Kosterlitz-Thouless on the XY model in the plane: what is a topological phase transition? (Nobel prize in physics 2016)
What is a spin glass? (Nobel prize in physics to Parisi 2021)
Követelmények szorgalmi időszakban:
Solving homework problems, oral presentation of two of them. A written midterm.
Konzultációs lehetőségek:
Jegyzet, tankönyv, felhasználható irodalom:
Rick Durrett: Essentials of stochastic processes, 2nd edition. Springer, 2011. https://services.math.duke.edu/~rtd/EOSP/EOSP2E.pdf
Rick Durrett: Random graph dynamics. Cambridge University Press, 2007. https://www.math.duke.edu/~rtd/RGD/RGD.pdf.
Rick Durrett. Probability: theory and examples. 5th edition. Cambridge University Press, 2019. https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf
Geoffrey Grimmett. Probability on graphs. Cambridge University Press, 2010.
Olle Haggstrom: Finite Markov Chains and Algorithmic Applications. Cambridge University Press, 2010
Gabor Pete: Probability and geometry on groups, http://math.bme.hu/~gabor/PGG.pdf
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