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Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time

Arthur Dremaux

TL;DR

The paper proves that for Brownian loop-soups on transient cable-graphs, conditioning on the total occupation time field Λ induces a singular conditional law for each individual loop relative to the unconditioned loop measure, for any loop-soup intensity c>0. The approach uses a detailed analysis of fast/quick points for Brownian motion and squared Bessel processes, and a deterministic lemma about how quick points propagate under sums, to connect local time features of Λ with the loop occupation times. The argument yields a robust, self-contained proof that leverages occupation-time densities and a loop-soup rewiring perspective, and is extended to a Brownian burglar scenario in one dimension via Ray–Knight-type decompositions. The results highlight a sharp, detectable discrepancy between conditioned and unconditioned loops from only local time information, without relying on Cameron–Martin-type drift arguments. Overall, the work connects fine path properties to global conditioning in loop-soup models and clarifies the one-dimensional nature of the singularity phenomenon.

Abstract

In this note, we show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field $Λ$: When conditioned on $Λ$, the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve $Λ$ impose an exceptional behaviour of all the loops when they go through these points.

Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time

TL;DR

The paper proves that for Brownian loop-soups on transient cable-graphs, conditioning on the total occupation time field Λ induces a singular conditional law for each individual loop relative to the unconditioned loop measure, for any loop-soup intensity c>0. The approach uses a detailed analysis of fast/quick points for Brownian motion and squared Bessel processes, and a deterministic lemma about how quick points propagate under sums, to connect local time features of Λ with the loop occupation times. The argument yields a robust, self-contained proof that leverages occupation-time densities and a loop-soup rewiring perspective, and is extended to a Brownian burglar scenario in one dimension via Ray–Knight-type decompositions. The results highlight a sharp, detectable discrepancy between conditioned and unconditioned loops from only local time information, without relying on Cameron–Martin-type drift arguments. Overall, the work connects fine path properties to global conditioning in loop-soup models and clarifies the one-dimensional nature of the singularity phenomenon.

Abstract

In this note, we show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field : When conditioned on , the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve impose an exceptional behaviour of all the loops when they go through these points.

Paper Structure

This paper contains 5 sections, 3 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fast points of Brownian motion
  3. "Quick points" of a squared Bessel process
  4. Proof of Proposition \ref{['Ploops']}
  5. Singularity of the Brownian burglar

Key Result

Proposition 1

When one conditions on the total occupation time field $\Lambda_c$, the conditional law of each individual loop in the loop-soup is singular with respect to that of an unconditioned loop (i.e., it is singular with respect to the loop measure).

Figures (1)

  • Figure 1: Sketch of the occupation time densities $\Lambda$ and $\ell_n$ of the whole loop-soup and of one loop $\beta_n$ in the loop-soup, on some edge of the cable-graph : Any $1$-quick point of $\Lambda$ in the support of $\ell_n$ is necessarily a quick point of $\ell_n$.

Theorems & Definitions (8)

  • Proposition 1
  • Lemma 2
  • proof
  • Remark 3
  • Remark 4
  • Proposition 5
  • proof
  • Remark 6