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Intensity doubling for Brownian loop-soups in high dimensions

Titus Lupu, Wendelin Werner

TL;DR

This work proves an intensity-doubling phenomenon for critical Brownian loop-soups on cable graphs in high dimensions ($d\ge 7$): macroscopic cycles within clusters either arise from a single macroscopic Brownian loop or from chains of small loops, with these two origins becoming asymptotically equally likely. The authors leverage the loop-soup switching property to construct a two-branch decomposition of macroscopic-cycle-containing clusters and show that, in the scaling limit, the cycles converge to a Brownian loop-soup with twice the usual intensity, reflected also in the Gaussian Free Field excursion sets. The analysis combines precise geometric control of large and mesoscopic loops, restricted-graph switching, and universal-cover couplings to establish the probabilistic dichotomy and its continuum limit. The results confirm a conjecture and illuminate how local loop-switching yields global scaling phenomena, with implications for GFF-based percolation and determinantal structures on high-dimensional graphs.

Abstract

We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^β$ when $β> 4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .

Intensity doubling for Brownian loop-soups in high dimensions

TL;DR

This work proves an intensity-doubling phenomenon for critical Brownian loop-soups on cable graphs in high dimensions (): macroscopic cycles within clusters either arise from a single macroscopic Brownian loop or from chains of small loops, with these two origins becoming asymptotically equally likely. The authors leverage the loop-soup switching property to construct a two-branch decomposition of macroscopic-cycle-containing clusters and show that, in the scaling limit, the cycles converge to a Brownian loop-soup with twice the usual intensity, reflected also in the Gaussian Free Field excursion sets. The analysis combines precise geometric control of large and mesoscopic loops, restricted-graph switching, and universal-cover couplings to establish the probabilistic dichotomy and its continuum limit. The results confirm a conjecture and illuminate how local loop-switching yields global scaling phenomena, with implications for GFF-based percolation and determinantal structures on high-dimensional graphs.

Abstract

We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of for that can be described as follows: In the box (and with a probability that goes to as goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than when is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than ) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .

Paper Structure

This paper contains 18 sections, 16 theorems, 9 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The setup and the more precise statement of the main result
  3. Preliminary estimates
  4. A first remark/warm-up
  5. The shape of large Brownian loops
  6. Different mesoscopic Brownian loops are in different clusters
  7. No connection between distant points on mesoscopic Brownian loops
  8. Summarizing
  9. Proof of the theorem
  10. A first use of the switching property
  11. Focusing on finitely many tubes suffices
  12. The switching property
  13. Using the switching property in a restricted graph
  14. Some direct consequences
  15. Statement of the key proposition and consequences
  16. ...and 3 more sections

Key Result

Theorem 1

For each $\varepsilon > 0$, the number of clusters in ${\mathcal{C}}(\varepsilon, N)$ remains tight when $N \to \infty$ and does converge in law to a Poisson random variable. Furthermore, for each $\beta > 4 / (d-2)$ and $\gamma > 2 / (d-4)$, one can find an explicit deterministic $u_N \to 0$ such t Furthermore, for each cluster $C$ in ${\mathcal{C}}(\varepsilon, N)$, one can define deterministica

Figures (7)

  • Figure 1: Each cluster containing a large cycle will be decomposed in one of these two ways with (almost) equal probability: Either the cycle is due to one big Brownian loop, or the cycle is created by a chain of very small Brownian loops.
  • Figure 2: An $(a,b,c)$ pinching loop
  • Figure 3: Low-probability configurations from Lemma \ref{['Lnotwo']} (two mesoscopic Brownian loops joined by a chain of loops) and Lemma \ref{['L2']} (a mesoscopic chain-of-loops-handle on top of a large Brownian loop)
  • Figure 4: The remaining case to exclude: The excursion of $B_k^2$ away from $B_k^1$ is almost all of $B_k^2$
  • Figure 5: The chain of small/mesoscopic loops from $u$ to $v$ -- if one loops is used twice at different windings (here in bold/red), there exists a chain of loops surrounding $\Delta_2$ (right)
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 1: Intensity doubling for $d \ge 7$
  • Remark 2
  • Lemma 3: No $(a,b,c)$ pinching, see Figure \ref{['fig:2']}
  • proof
  • Lemma 4: No two mesoscopic Brownian loops in the same cluster, see the left part of Figure \ref{['fig:3']}
  • proof
  • Lemma 5: No connections between distant points on mesoscopic Brownian loops, see the right part of Figure \ref{['fig:3']}
  • proof
  • Proposition 6: The loop version of the switching property -- Corollary 5 from W3
  • Lemma 7
  • ...and 18 more