Selected References on Universal Objects
Overview
Continuum Random Tree References
- Duquesne, and Le Gall. “Random Trees, Levy processes, and Spatial Branching Processes.” (PDF)
- Lalley. “Levy Processes, Stable Processes, and Subordinators.” (PDF)
- Lawler. “Bessel Processes.” (PDF)
- Revuz, and Yor. Continuous Martingales and Brownian Motion. Springer, 2004. ISBN: 9783540643258.
- Bertoin. Levy Processes. Cambridge University Press, 1998. ISBN: 9780521646321.
- Johnson. “Poisson Point Proccesses.” (PDF - 2.5MB)
- Curien, and Kortchemski. “Stable Loop Trees.” Electronic Journal of Probability 19, no. 108 (2014): 1–35.
- Aldous. “Continuum Random Tree I.” The Annals of Probability 19, no. 1 (1991): 1–28.
- ———. “Continuum Random Tree II.” (PDF)
- ———. “The Continuum Random Tree III.” The Annals of Probability 21, no. 1 (1993): 248–89.
- Mörters, and Peres. “Brownian Motion.” (PDF - 7.2MB)
- Timorin. “Moore’s Theorem.” (PDF) 2012.
Random Planar Maps and the Brownian Map References
- Le Gall, and Miermont. “Scaling Limits of Random Trees and Planar Maps.” 2011.
- Le Gall. “Random Geometry on the Sphere.” 2014.
- Bernardi. “Slides on Cori-Vauquelin-Schaeffer Bijection.” (PDF - 1.5MB) and “Brownian Map Convergence.” (PDF - 1.7MB)
- Sheffield. “Quantum Gravity and Inventory Accumulation.” 2011.
- Miller, Sheffield. “An Axiomatic Characterization of the Brownian Map.” (PDF - 1.2MB) 2015.
Gaussian Free Field References
- Sheffield. “Gaussian Free Fields for Mathematicians.” 2003.
- Werner. “Topics on the Two-dimensional Gaussian Free Field.” (PDF) 2014.
- Duplantier, Rhodes, et al. “Log-correlated Free Field in General Dimension.” (PDF) 2014.
- Lodhia, Sheffield, et al. “Fractional Gaussian Fields: A Survey.” (PDF - 3.3MB) 2016.
Liouville Quantum Gravity References
- Duplantier, and Sheffield. “Liouville Quantum Gravity and KPZ.” 2010.
- Garban. “Quantum Gravity and the KPZ Formula.” 2012.
- Berestycki. “Introduction to the Gaussian Free Field and Liouville Quantum Gravity.”
Schramm-Loewner Evolution and Discrete Analogs References
- Werner. “Random Planar Curves and Schramm-Loewner Evolutions.” 2003.
- “Conformally Invariant Processes in the Plane: Book.” (PS - 1.7MB) (Lawler—save and use online ps2pdf if your machine doesn’t have postscript).
- Kager, and Bernard. “A Guide to Stochastic Loewner Evolution and its Applications.” Journal of Statistical Physics 115, no. 5 (2004): 1149–229.
- Berestycki, N., and J. R. Norris. “Lectures on Schramm-Loewner Evolution.” (PDF)
Growth Models References
- Witten, and Sander. “Diffusion Limited Aggregation.” (PDF - 1.7MB) The American Physical Society 27, no. 9 (1983).
- Kesten. “DLA Bounds.” Stochastic Processes and their Applications 25 (1987): 165–84.
- Niemeyer, Pietronero, et al. “Dielectric Breakdown Model.” (PDF) Physical Review Letter 52, no. 12 (1984).
- Quaste. “Introduction to KPZ.” (PDF) Current Development in Mathematics, 2011.
- Corwin, Quastel, et al. “Renormalization Fixed Point of the KPZ Universality Class.” (PDF)
- Corwin. “The Kardar–Parisi–Zhang Equation and Universality Class.” Random Matrices: Theory and Applications 1, no. 1 (2012).
- Borowin, Corwin, et al. “Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension.” Communications on Pure and Applied Mathematics 67, no. 7 (2014): 1129–214.
Yang Mills References
- Jaffe, and Witten. “Quantum Yang-Mills Theory.” (PDF)
- Chatterjee. “Large N Lattice Gauge Theory.” (PDF) 2016.
- Levy. “Large N Master Field in 2D.” 2012.
Selected References on Universal Object Relationships
GFF + SLE
- Miller, and Sheffield. “Imaginary Geometry I: Interacting SLEs.” 2012. (See also parts II, III, IV)
- Schramm, and Sheffield. “A Contour Line of the Continuum Gaussian Free Field.” 2010.
LQG + LQG = LQG + SLE
Sheffield. “Conformal Weldings of Random Surfaces: SLE and the Quantum Gravity Zipper.” 2015.
CRT + CRT = LQG + SLE
Miller, and Sheffield. “Liouville Quantum Gravity as a Mating of Trees.” 2014.
LQG + Reshuffled SLE = LQG + DBM
Miller, and Sheffield. “Quantum Loewner Evolution.” 2013.
LQG = TBM
Miller, and Sheffield. “Liouville Quantum Gravity and the Brownian Map.” (PDF)
