Conformally invariant scaling limits of random curves and correlations

Alex Karrila

Research output: ThesisDoctoral ThesisCollection of Articles


This thesis studies scaling limits of critical random models on planar graphs, when a fine-mesh graph approximates a planar domain. Such studies are motivated by quantum field theoretic predictions that suggest the emergence of intricate, conformally invariant structures from simple combinatorial models. We take a mathematical approach, formalizing our results in terms of SLE type conformally invariant random curves or certain expected values, called boundary correlation functions in physics.  In the first publication of this thesis, we study two related random models, the uniform spanning tree (UST) and the loop-erased random walk (LERW). We obtain conformally covariant expressions for the scaling limit probabilities of certain UST branch connectivities and of LERW boundary visits. These expressions are solutions to partial differential equations (PDEs) of second and third order, respectively, and such solutions appear in Conformal field theory (CFT) as boundary correlation functions. CFT predicts such PDEs of arbitrarily high order, and this is among the first verifications of higher-than-second order PDEs.  The PDE solutions from the first publication can also be interpreted as weights that conjecturally convert the SLE(2) random curve measure, the scaling limit of a UST branch and a LERW, to multiple or boundary-visiting SLE(2). In the second publication, we elaborate this connection by finding an explicit relation between certain multiple SLE(k) weights, called pure partition functions, and certain CFT boundary correlation functions, called conformal blocks.  The third and fourth publication concern the weak convergence of the joint law of multiple lattice curves to multiple SLE type random curves. We first provide a result that guarantees at least subsequential convergence to some limiting random curves, given certain standard crossing estimates in the lattice models. Second, we show how such limits can be described by iteratively sampling the curves one by one from the weighted one-curve SLE(k) measures described above. These tools are applied to characterize the scaling limits of multiple curves in various random models, such as multiple UST branches.
Translated title of the contributionSatunnaiskäyrien ja korrelaatioiden konformi-invariantteja skaalausrajoja
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Kytölä, Kalle, Supervising Professor
Print ISBNs978-952-60-8631-6
Electronic ISBNs978-952-60-8632-3
Publication statusPublished - 2019
MoE publication typeG5 Doctoral dissertation (article)


  • scaling limits
  • lattice models
  • random curves
  • SLE
  • conformal field theory
  • loop-erased random walk (LERW)

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