Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Planar UST Branches and $c=-2$ Degenerate Boundary Correlations

Alex Karrila, Augustin Lafay, Eveliina Peltola, Julien Roussillon

TL;DR

This work provides a rigorous CFT framework for boundary effects in the wired UST at central charge $c=-2$, showing that all boundary-to-boundary connection probabilities converge in the scaling limit to explicit determinants that serve as conformal blocks for degenerate boundary fields. It proves higher-order BPZ PDEs, reveals a valenced Temperley-Lieb algebra action on boundary correlators, and establishes a precise fusion/covariance structure linking discrete probabilistic models to CFT data. The methods combine probabilistic, combinatorial, and representation-theoretic techniques to derive scaling limits, fusion rules, and asymptotics for fused boundary observables, bridging lattice models, SLE, and degenerate CFT blocks. These results provide a broad dictionary between planar UST boundary phenomena and $c=-2$ CFT, with potential implications for monodromy-invariant bulk correlators and fused-SLE descriptions of multiple boundary interfaces.

Abstract

We provide a conformal field theory (CFT) description of the probabilistic model of boundary effects in the wired uniform spanning tree (UST) and its algebraic content, concerning the entire first row of the Kac table with central charge $c=-2$. Namely, we prove that all boundary-to-boundary connection probabilities for (potentially fused) branches in the wired UST converge in the scaling limit to explicit CFT quantities, expressed in terms of determinants, which can also be viewed as conformal blocks of degenerate primary fields in a boundary CFT with central charge $c=-2$. Moreover, we verify that the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs (i.e., Virasoro degeneracies) of arbitrary orders hold, and we also reveal an underlying valenced Temperley-Lieb algebra action on the space of boundary correlation functions of primary fields in this model. To prove these results, we combine probabilistic techniques with representation theory.

Planar UST Branches and $c=-2$ Degenerate Boundary Correlations

TL;DR

This work provides a rigorous CFT framework for boundary effects in the wired UST at central charge , showing that all boundary-to-boundary connection probabilities converge in the scaling limit to explicit determinants that serve as conformal blocks for degenerate boundary fields. It proves higher-order BPZ PDEs, reveals a valenced Temperley-Lieb algebra action on boundary correlators, and establishes a precise fusion/covariance structure linking discrete probabilistic models to CFT data. The methods combine probabilistic, combinatorial, and representation-theoretic techniques to derive scaling limits, fusion rules, and asymptotics for fused boundary observables, bridging lattice models, SLE, and degenerate CFT blocks. These results provide a broad dictionary between planar UST boundary phenomena and CFT, with potential implications for monodromy-invariant bulk correlators and fused-SLE descriptions of multiple boundary interfaces.

Abstract

We provide a conformal field theory (CFT) description of the probabilistic model of boundary effects in the wired uniform spanning tree (UST) and its algebraic content, concerning the entire first row of the Kac table with central charge . Namely, we prove that all boundary-to-boundary connection probabilities for (potentially fused) branches in the wired UST converge in the scaling limit to explicit CFT quantities, expressed in terms of determinants, which can also be viewed as conformal blocks of degenerate primary fields in a boundary CFT with central charge . Moreover, we verify that the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs (i.e., Virasoro degeneracies) of arbitrary orders hold, and we also reveal an underlying valenced Temperley-Lieb algebra action on the space of boundary correlation functions of primary fields in this model. To prove these results, we combine probabilistic techniques with representation theory.

Paper Structure

This paper contains 43 sections, 33 theorems, 204 equations, 7 figures.

Table of Contents

  1. Introduction
  2. CFT framework
  3. CFT consequences
  4. UST framework
  5. Main results: UST connection probabilities and boundary correlations
  6. Precise scaling limit setup
  7. Convergence of connection probabilities
  8. Degenerate boundary correlation functions
  9. Algebraic results: Temperley-Lieb representations
  10. Discrete connection probabilities and Fomin type formulas
  11. General inverse Fomin type sums
  12. Discrete connection probabilities without fusion
  13. Fusion of discrete connection probabilities
  14. Scaling limit results
  15. Scaling limits of connection probabilities: Proof of Theorem \ref{['thm:scaling limit of pinched pertition functions']}
  16. ...and 28 more sections

Key Result

Theorem 1.2

Let admissible $2N$-polygons $(\Lambda^\delta; e_1^\delta, \ldots, e_{2N}^\delta)$ converge as $\delta \to 0$ to an admissible $d$-polygon $(\Lambda; p_1, \ldots, p_d)$ with valences $\varsigma = (s_1,\ldots,s_d)$ in the Carathéodory sense, furthermore so that the fused boundary edges are for each $ where is a Fomin type determinant (defined in Section subsec:inv Fomin); the kernel $\mathfrak{K}$

Figures (7)

  • Figure 1.1: Two simulations of the wired uniform spanning tree (UST) on small graphs, illustrating the central combinatorial notions to this work. The left panel depicts a wired uniform spanning tree on a grid graph of $20 \times 20$ squares. The right panel depicts the same model on $30 \times 30$ squares, conditional on the event $\mathrm{Conn}$ with the marked boundary edges $e_1, \ldots, e_8$, where the related boundary-to-boundary UST branches are highlighted in (shades of) red. In this case, the random connectivity of the interfaces of interest is $\vartheta_{\mathrm{UST}} = \{ \{ 1, 4 \}, \{ 2, 3\}, \{ 5, 6 \}, \{ 7, 8 \} \}$.
  • Figure 1.2: Simulations of USTs with some boundary branches highlighted.
  • Figure 1.3: An illustration of the the iterated limits that by Theorem \ref{['thm:CFT properties']}(FUS) produce $\mathcal{Z}_\alpha$ with $"$increasingly fused$"$ valenced patterns $\alpha$, all corresponding to the same $"$unfused pattern$"$$\imath(\alpha)$. Parts (PDE), (POS) and (LIN) of Theorem \ref{['thm:CFT properties']} are proven inductively over such increasing fusion.
  • Figure 6.1: Link pattern $\hat{\alpha}$ obtained from $\alpha$ by fusing two endpoints.
  • Figure 6.2: Example link patterns $\alpha$ in the two special cases $\mathrm{m}_{j,j+1} =0$ (left) and $\mathrm{m}_{j,j+1} =\min(s_{j},s_{j+1})$ (right), which lay the basis for the proof of Theorem \ref{['thm:ASY']}, and for which $C(s_j, s_{j+1}, \mathrm{m}_{j,j+1}) > 0$ is obtained explicitly.
  • ...and 2 more figures

Theorems & Definitions (71)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem A: See KKP:Boundary_correlations_in_planar_LERW_and_UST for this formulation
  • Remark 2.3
  • Proposition 2.4
  • ...and 61 more