Statistical Mechanics of Lattice Systems

V5B7 - Advanced Topics in Analysis - WS 2023/24

Tuesday 14-16 and Wednesday 12-14, Seminarraum S.006 (Endenicher Alle 60)

Instructors: Luca Fresta, Paolo Rinaldi

Basis link

Reference: S. Friedli, Y. Velenik. Statistical Mechanics of Lattice Systems, Cambridge University Press 2017 (link)

Course Diary:

  • 22/11: Introduction to the second part of the course, examples of other lattice models (Sec. 6.1). Probability space for lattice systems:  canonical projections, product sigma-algebra, product topology, local observables, states&probability measures (parts of Sec. 6.2, complemented with Kallenberg, Foundations of modern probability.)
  • 28/11: Infinite-volume measures and Kolmogorov's extension theorem (parts of Sec. 6.2, Sec. 6.12.1, 6.12.3 and parts of App. B.5.1). DLR formalism: interactions, Hamiltonian, Gibbsian specifications (parts of Sec. 6.3.2 and 6.10.1)
  • 29/11: properties of Gibbsian specifications, DLR definition of infinite-volume measure  (Sec. 6.3.2, parts of Sec. 6.3 specialising to Gibbsian specifications). Introduction to GFF, definition of the model and overview (Sec. 8,8.1, and 8.1.1)
  • 05/12: Gaussian vectors and fields (Sec. 8.2, parts of App. B.9), lattice Laplacian, lattice Green's identities (Sec. 8.3)
  • 12/12: Dirichlet problem, inverse Laplacian, Gaussian density for the massless GFF (parts of Sec. 8.3 and parts of Sec. 8.4)
  • 13/12: Random-walk representation of mean and covariance of the massless GFF (parts of Sec. 8.4 and parts of App. B.13)
  • 19/12: Recurrence and transience (Def. B.71, Theorem B.72, Corollary B.73), applications to the infinite-volume covariance (end of Sec. 8.4.1, some technical results scattered through Sec. 8.4.2, Theorem B.76), absence of infinite-volume massless GFF in d=1,2 (Theorem 8.19)
  • 20/12: Conditional expectation and basic properties (Lemma B.49, Lemma B.50), specifications as conditional expectations (Sec. 6.3.1), random-walk representation of the conditional expectation of the field (Lemma 8.24), existence of infinitely many infinite-volume massless GFF in d=>3 (Theorem 8.21)
  • 16/01: Results on the massive GFF (Secs. 8.5.1 and 8.5.2), definition and overview of O(N)-symmetric models (Secs. 9.1 and 9.1.1), compactness of the space of prob. measures (Sec. 6.4.2)
  • 17/01: Feller property and existence of the infinite-volume measure (Sec. 6.4.3 untile Th. 6.26), statment of the Mermin--Wagner theorem and some consequences (parts of Sec. 9.2), quantitative statement in finite volume (Proposition 9.7)
  • 23/01: Relative entropy and Pinsker's inequality (Secs. B.12.1 and B.12.2) and first part of the proof of Proposition 9.7
  • 24/01: end of the proof of Proposition 9.7, by direct Ansatz and by minimisation of the Dirichlet energy (Sec. 9.2.2 and Theorem B.74)

Oral exams: first session on Wednesday 7 and Thursday 8, February 2024. Second session in the week 18-22.03.24. 

List of topics for the exam: link.

 

 

 

 

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