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arxiv: 2604.13387 · v1 · submitted 2026-04-15 · 🧮 math.PR · math-ph· math.MP

Multiradial Schramm-Loewner evolution: Infinite-time large deviations and transience

Osama Abuzaid , Vivian Olsiewski Healey , Eveliina Peltola This is my paper

Pith reviewed 2026-05-10 13:03 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords multiradial Schramm-Loewner evolutionlarge deviation principleLoewner energytransiencecommon capacity parameterizationradial multichordsBrownian loop measureVirasoro cocycle
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The pith

Multiradial SLE curves satisfy an infinite-time large deviation principle with the multiradial Loewner energy as rate function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a finite-time large deviation principle for multiradial Schramm-Loewner evolution to infinite time. It establishes that as kappa tends to zero the random curves, when parameterized by common capacity, obey a large deviation principle whose good rate function is the multiradial Loewner energy. The argument rests on uniform escape probability estimates that generalize single-curve bounds and pass to the limit. A direct consequence is that the curves are transient at their common terminal point whenever kappa is at most 8/3. The same limit also supplies explicit linear asymptotics for the Brownian loop measure interaction term in finite-energy radial multichords.

Core claim

We prove that the family of multiradial SLE(kappa) measures satisfies a large deviation principle as kappa tends to zero in the topology of common-capacity-parameterized curves, with good rate function given by the multiradial Loewner energy. This extends the earlier finite-time Hausdorff-metric result and yields transience of the curves for kappa less than or equal to 8/3 at the common terminal point. As a corollary we obtain explicit asymptotics for the Brownian loop measure interaction term, linear in capacity time and coinciding with a chosen cocycle for the Virasoro algebra.

What carries the argument

The multiradial Loewner energy, which quantifies the exponential cost of deviations from deterministic limit shapes in the common-capacity parameterization of several curves growing simultaneously from a single point.

If this is right

  • Multiradial SLE(kappa) curves with kappa at most 8/3 remain transient at their common terminal point.
  • The Brownian loop measure interaction term for finite-energy radial multichords grows linearly with capacity time.
  • The large deviation principle holds in the common-capacity topology rather than the Hausdorff metric used for finite time.
  • The escape estimates streamline the finite-time proof by providing uniform control across multiple curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The transience statement extends single-curve results to the simultaneous growth of several curves sharing a terminal point.
  • The explicit asymptotics for the loop-measure interaction term supply a probabilistic construction of a Virasoro cocycle that may be compared with algebraic definitions.
  • The same escape-estimate technique could be tested on other variants of Loewner evolution that admit a common-capacity parameterization.

Load-bearing premise

The escape probability estimates for multiradial SLE(kappa) in the common parameterization extend the single-curve estimates and hold uniformly enough to pass to the infinite-time limit.

What would settle it

Numerical sampling of multiradial SLE paths for small kappa that shows the probability of staying near a non-energy-minimizing configuration up to large common capacity time fails to decay at the exponential rate given by the multiradial Loewner energy.

read the original abstract

In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(\kappa)$, as $\kappa \to 0$, with good rate function being the multiradial Loewner energy. Here, we extend this result to infinite time in the topology of common-capacity-parameterized curves, and streamline the proof. A main step is to derive detailed escape probability estimates for multiradial SLE$(\kappa)$ curves in the common parameterization, which extend the single-curve estimates achieved in [AP26]. As a by-product, we also get that multiradial SLE$(\kappa)$ curves, with $\kappa \leq 8/3$, are transient at their common terminal point, generalizing [FL15, HL21]. As a corollary to the LDP result, we obtain explicit asymptotics of the Brownian loop measure interaction term for finite-energy radial multichords, which is linear in the capacity-time and coincides with a certain choice of a cocycle for the Virasoro algebra.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper extends the finite-time large deviation principle in the Hausdorff metric for multiradial SLE(κ) as κ→0, established in prior work, to an infinite-time LDP in the topology of common-capacity-parameterized curves, with good rate function given by the multiradial Loewner energy. A central step is the derivation of detailed escape probability estimates for multiradial SLE(κ) in the common parameterization that extend single-curve bounds. As by-products, the authors obtain transience of multiradial SLE(κ) curves (κ≤8/3) at their common terminal point and explicit linear-in-capacity-time asymptotics for the Brownian loop measure interaction term of finite-energy radial multichords, coinciding with a Virasoro cocycle choice.

Significance. If the uniformity of the escape estimates holds, the result is significant: it provides the first infinite-time LDP for multiradial SLE in a natural topology, generalizes known transience statements, and yields concrete asymptotics linking to conformal field theory structures. The use of the deterministic multiradial Loewner energy as rate function and the streamlining of the finite-time argument are strengths; the work supplies reproducible predictions via the explicit rate function and interaction asymptotics.

major comments (2)
  1. [derivation of escape probability estimates] The escape probability estimates (main step highlighted in the abstract) must supply constants uniform in the number of curves, their mutual positions, and the common terminal point to justify passage to the infinite-time limit in the common-capacity topology. If radial interactions introduce configuration-dependent corrections that grow with capacity time or arm count, the large-deviation upper and lower bounds may fail to close even if they hold at each finite time; explicit uniformity statements or bounds independent of these parameters are required.
  2. [transience by-product] The transience claim for κ≤8/3 at the common terminal point is presented as a by-product of the escape estimates. It is necessary to confirm that the estimates yield a uniform positive probability of escape from any neighborhood of the terminal point that is independent of the number of arms and their initial configuration, without additional assumptions on the common terminal point.
minor comments (1)
  1. [introduction] The abstract cites [AHP24], [AP26], [FL15, HL21] but the introduction should explicitly state how the new escape estimates improve upon or differ from the single-curve constants in [AP26].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below and will revise the manuscript to make the required uniformity statements explicit.

read point-by-point responses

  1. Referee: The escape probability estimates (main step highlighted in the abstract) must supply constants uniform in the number of curves, their mutual positions, and the common terminal point to justify passage to the infinite-time limit in the common-capacity topology. If radial interactions introduce configuration-dependent corrections that grow with capacity time or arm count, the large-deviation upper and lower bounds may fail to close even if they hold at each finite time; explicit uniformity statements or bounds independent of these parameters are required.

    Authors: We agree that uniformity of constants is essential for closing the infinite-time LDP. The escape estimates in the paper are obtained by extending the single-curve bounds of [AP26] via the multiradial Loewner equation in common capacity parameterization; the interaction terms are controlled by deterministic energy bounds that yield constants depending only on κ and the capacity horizon, independent of the number of arms, their relative positions, and the choice of common terminal point. No growing configuration-dependent corrections appear. We will insert an explicit uniformity proposition (or remark) in the revised Section 3 stating these independence properties. revision: yes

  2. Referee: The transience claim for κ≤8/3 at the common terminal point is presented as a by-product of the escape estimates. It is necessary to confirm that the estimates yield a uniform positive probability of escape from any neighborhood of the terminal point that is independent of the number of arms and their initial configuration, without additional assumptions on the common terminal point.

    Authors: The transience statement is obtained by summing the escape probabilities over successive capacity intervals; the uniformity already established in the estimates ensures the lower bound on the escape probability from any fixed neighborhood is positive and independent of arm count and initial configuration. The common terminal point affects only the deterministic starting data at time zero and enters no further assumptions. We will revise the transience corollary to record this uniformity explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension relies on independent escape estimates

full rationale

The paper cites prior work [AHP24] for the finite-time LDP and [AP26] for single-curve estimates, but the central extension to infinite time in the common-capacity topology is achieved by deriving new detailed escape probability estimates for multiradial SLE(κ) that are claimed to extend the single-curve bounds uniformly. The good rate function is the multiradial Loewner energy taken from prior deterministic theory, not redefined or fitted within this paper. The transience result is presented as a by-product generalizing external references [FL15, HL21]. No derivation step reduces by construction to its inputs, no fitted quantity is relabeled as a prediction, and no uniqueness theorem or ansatz is smuggled via self-citation in a load-bearing way. The argument chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the established existence and basic properties of multiradial SLE processes and Loewner chains from prior literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Existence, uniqueness, and basic conformal invariance properties of multiradial SLE(kappa) processes
    Invoked implicitly when stating the curves and their parameterization; taken from the cited prior work on SLE.

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