Random Processes in Conformal Field Theory
Summary
Random processes are central to contemporary approaches in two-dimensional conformal field theory, where they furnish a bridge between probabilistic models and quantum fields endowed with conformal invariance. The Schramm–Loewner evolution (SLE) has emerged as a canonical family of random curves describing scaling limits of interfaces in critical lattice systems. Parallel developments in the theory of the Gaussian free field (GFF) have revealed that its flow lines encode random geometries whose local behaviour is governed by SLE trajectories. In tandem, Gaussian multiplicative chaos provides a rigorous construction of random measures that realise Liouville quantum gravity (LQG), endowing planar domains with a conformally invariant random metric. These tools have unlocked quantitative descriptions of fractal boundaries, multifractal spectra and metric structures in critical phenomena. At the same time, novel growth processes such as quantum Loewner evolution (QLE) offer a dynamic perspective on how random surfaces evolve under conformal welding. Together, these random processes have deepened our understanding of universality classes in statistical mechanics, the geometry of quantum gravity in two dimensions and the interplay between discrete models and their continuum limits.
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Random Processes in Conformal Field Theory publication trend
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Technical terms
Conformal Field Theory (CFT): A quantum field theory invariant under angle-preserving (conformal) transformations, often describing critical phenomena in two dimensions.
Schramm–Loewner Evolution (SLE): A one-parameter family of random fractal curves defined via a conformally invariant stochastic differential equation, encoding scaling limits of critical interfaces.
Gaussian Free Field (GFF): A random generalised function on a domain whose covariance is the Green’s function of the Laplacian, serving as a continuum analogue of discrete height models.
Gaussian Multiplicative Chaos (GMC): A method to construct random measures by exponentiating a log-correlated Gaussian field, central to rigorous formulations of Liouville quantum gravity.
Liouville Quantum Gravity (LQG): A random geometry obtained by equipping a two-dimensional domain with a metric derived from exponentiating the Gaussian free field, modelling a quantised surface.
Quantum Loewner Evolution (QLE): A growth process on LQG surfaces that defines a random metric via the amount of “quantum time” required to link two points, yielding a metric geometry on random surfaces.
References
- Imaginary geometry I: interacting SLEs. Probability Theory and Related Fields (2016).
- Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probability Theory and Related Fields (2017).
- SLE and the free field: Partition functions and couplings. Journal of the American Mathematical Society (2009).
- Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric. Inventiones Mathematicae (2019).
- Gaussian multiplicative chaos and applications: A review. Probability Surveys (2014).
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