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A switching identity for cable-graph loop soups and Gaussian free fields

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Conditioning two points on a cable graph to belong to the same Brownian loop-soup cluster is the same, in law, as adding an odd number of independent Brownian excursions to an unconditioned loop-soup.

desk verdict Genuinely new switching identity with strong consequences; Section 3 has a likely factor-of-two normalization slip, but the theorem is probably true and deserves serious review. read the letter →

arxiv 2502.06754 v4 pith:JMAOOGTX submitted 2025-02-10 math.PR math-phmath.MP

Wendelin Werner This is my paper · ORCID
classification math.PRmath-phmath.MP MSC 60J6560K35
keywords Brownianloop-soupcablegraphGaussianfreefieldswitchingidentitypercolationincipientinfiniteclusterrandomevensubgraphsMarkovloops
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a switching identity for critical Brownian loop-soups on cable graphs, equivalently for the square of the Gaussian free field. The claim is that conditioning two points x and y to lie in the same loop cluster is, in law, the same as taking an unconditioned loop-soup and overlaying an odd number of independent Brownian excursions joining x and y. The identity holds at the level of occupation-time fields, not just cluster topology, and extends to the boundary case where the two points are on the boundary of the same cluster, where a single excursion suffices. A reader should care because it converts a difficult conditional object into a sum of simple independent pieces, giving explicit laws for incipient infinite clusters, simplifications of recent results, and new tools for multi-point and high-dimensional questions.

What carries the argument

The central object is the Brownian loop-soup, a Poisson point process of unrooted Brownian loops on a cable graph (a metric graph whose edges are segments), whose occupation-time field has the same law as the square of the cable-graph Gaussian free field. The load-bearing mechanism is the parity lemma: conditionally on $\Lambda(x)=a^2$ and $\Lambda(y)=b^2$, the loops meeting $\{x,y\}$ decompose into excursions, and the number of excursions joining $x$ to $y$ is a Poisson random variable conditioned to be even. Comparing this even description with the unconditioned Poisson description, and reweighting by the event that the two signs of the GFF agree, gives the switching: conditioning on $x\leftrightarrow y$ makes the number of joining excursions odd. The paper supplies three proofs of this mechanism: one via discrete Markov-chain approximations, one via Laplace transforms in the spirit of Dynkin's isomorphism, and one via random even subgraphs in the Ising random current representation.

What would settle it

On a small cable graph such as a single edge with endpoints $x$ and $y$, compute the conditional Laplace transform of the occupation-time field given $x\leftrightarrow y$, $\Lambda(x)=a^2$ and $\Lambda(y)=b^2$; the switching identity requires it to factor as the product of the Laplace transforms of an unconditioned loop-soup in the interior, two Poisson excursion processes away from the endpoints, and an odd-conditioned Poisson process of joining excursions. If the odd-conditioned factor does not appear with the exact intensity $ab\nu_{x,y}$, or if the four factors are not independent, the identity is false; the one-dimensional single-edge case corresponds to a classical Bessel bridge decomposition, so a discrepancy there would falsify the general theorem.

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Extended reading notes

Core claim

Theorem 2 states that, conditionally on $x \leftrightarrow y$, $\Lambda(x)=a^2$ and $\Lambda(y)=b^2$, the critical loop-soup occupation time $\Lambda=\Gamma^2$ has the same law as the sum of the occupation times of four independent inputs: an unconditioned critical loop-soup in $G\setminus\{x,y\}$; a Poisson process of excursions away from $x$ with intensity $a^2$ times the excursion measure; a Poisson process of excursions away from $y$ with intensity $b^2$ times the excursion measure; and a Poisson process of excursions joining $x$ and $y$ with intensity $ab$ times the excursion measure, conditioned so that the number of joining excursions is odd. The boundary version (Theorem 1) says that conditioning two points to be on the boundary of the same cluster is equivalent to overlaying one independent Brownian excursion between them. As a consequence, the incipient infinite cluster measure in $\mathbb{Z}^d$ for $d\ge 3$ exists and is described by an unconditioned loop-soup reweighed by the square root of its local time at the origin plus one independent Brownian excursion from the origin to infinity (Theorem 3).

Load-bearing premise

The most exposed premise is that the fine discrete approximations of the cable-graph loop-soup converge to the continuous process in the way needed for the parity lemma to pass to the limit; the Laplace-transform proof in Section 3 provides an independent route that does not rely on that approximation.

Editorial extensions

If this is right

  • The incipient infinite cluster of the loop-soup in $\mathbb{Z}^d$ for $d\ge 3$ exists and has the explicit description of a loop-soup reweighed by the square root of its local time at the origin, plus one independent Brownian excursion from the origin to infinity.
  • Conditioned versions of the incipient infinite cluster have the same form: fixing a boundary point, or fixing the occupation time at the origin, yields the overlay of an unconditioned or conditioned loop-soup with one excursion to infinity.
  • For a loop-soup superposed on a Brownian interlacement, conditioning the origin to be connected to infinity gives an odd number of Brownian excursions from the origin to infinity, with the interlacement conditioned to avoid the origin.
  • Conditionally on the occupation-time field, the parity of loop crossings forms a uniform random even subgraph, which yields independent fair coins for the winding parity of clusters and proves the intensity doubling conjecture for loop-soups in dimensions $d\ge 7$.
  • The switching property gives new upper and lower bounds for multi-point connection probabilities and for the size of the largest clusters in dimensions $d=3,4,5$, complementing recent results obtained by renormalization arguments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the switching identity survives to the continuum scaling limit, then in dimensions $d=3,4,5$ the continuum loop-soup clusters would inherit an even simpler conditional law: conditioning two points to be connected would amount to adding a single Brownian excursion, as in the two-dimensional results cited in the paper.
  • The explicit overlay description of the incipient infinite cluster suggests that spectral properties of the cluster (e.g., Alexander–Orbach type exponents) could be studied by analysing a Brownian excursion in the random environment created by an unconditioned loop-soup.
  • A finite-graph Monte Carlo check could test the parity mechanism directly: sample loop-soups conditioned on $\Lambda(x)=a^2$, $\Lambda(y)=b^2$ and $x\leftrightarrow y$, and count the number of independent excursion bridges joining $x$ and $y$; the switching identity predicts a Poisson law with mean $ab$ times the mass of $\nu_{x,y}$, conditioned to be odd.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper states and proves a "switching identity" for critical Brownian loop-soups and the Gaussian free field on cable graphs. The main statement, Theorem 2, says that conditionally on two points x and y being in the same loop-soup cluster and on the occupation times at x and y taking values a^2 and b^2, the occupation field has the law of the sum of four independent inputs: a loop-soup avoiding x and y, Poisson processes of excursions away from x and away from y with intensities proportional to a^2 and b^2, and a Poisson process of excursions joining x and y whose total number is conditioned to be odd. The paper gives three routes to this result: a discrete Markov-chain parity lemma, a Laplace-transform/Dynkin computation, and a random-current-style switching along circuits. It then derives consequences: a boundary-point version (Theorem 1), an incipient-infinite-cluster measure (Theorem 3), interlacement versions, parity identities for windings of loop-soup clusters, and multiple-point estimates for loop-soup percolation.

Significance. If Theorem 2 is correct, it is a genuinely striking and sharp description: conditioning on a connection is exactly an odd-path insertion into an unconditioned configuration. The statement is clean, falsifiable, and has direct consequences that are not accessible by prior methods, including a simple construction of the IIC measure in all d >= 3 and new parity identities for loop-soup clusters. The paper also correctly connects the result to the rewiring property of [66], to random-current switching in the Ising model, and to Pitman--Yor decompositions of Bessel bridges. The multiple proof strategies are a strength, and the consequences in Sections 5--6 are appropriately stated as applications. However, as written, the Laplace-transform proof in Section 3 contains a load-bearing Gaussian computation error, and the first proof in Section 2 relies on a parity lemma whose convergence argument is only sketched. The central claim is likely correct, but the manuscript needs a corrected and/or completed proof before it can be accepted.

major comments (2)
  1. [Section 3, displayed formula after "By inspecting the variance..."] For a centered Gaussian field Gamma^k with covariance G^k, E[exp(-2 int Gamma^k(x) k(x) Phi(x) dx)] equals exp(2 int int G^k(x,y) k(x) k(y) Phi(x) Phi(y) dx dy), not exp(int int G^k kk Phi Phi) as written. Moreover, under the paper's own normalization (Cov(Gamma)=G and Lambda=Gamma^2), the field obtained by reweighting with exp(-int Gamma_0^2 k) has covariance (G^{-1}+2k)^{-1}, which is the Green function with killing rate 2k, not k, in the usual Brownian convention. The identification of the cross term (3) with the Laplace transform of the Poisson process of excursions joining x1 and x2 relies on this computation, so the Section 3 proof of the parity lemma and of the switching property is not valid as written. Please correct the factor and state the normalization of G^k explicitly, or restructure Section 3 so that the claimed identity is derived from a correct Gaussian calculation.
  2. [Section 2, Lemma 8 and its proof] The passage from Formula (2) to the claimed even-Poisson law is not justified. Substituting A ~ a^2 K, B ~ b^2 K and p(x,y)=alpha/K in (2) gives, for a jump count 2t, weights proportional to (alpha a b)^{2t}/t! (up to factors independent of t), whereas a Poisson variable conditioned to be even has weights proportional to mu^{2t}/(2t)!. The displayed asymptotics therefore do not identify the limiting conditional distribution. The bracketing argument with P_1 <= N <= P_2 also does not by itself prove the conditional law. Since Lemma 6 and hence the first proof of Theorem 2 depend on this step, the Section 2 route needs a complete proof or a precise reference to a result that contains this convergence.
minor comments (4)
  1. [Section 3, proof of the switching property] In the displayed computation of E[1_{x1 connected to x2} exp(-int Gamma^2 k) | ...], the denominator e^m - e^{-m} should be e^m + e^{-m}; as printed, the equality to sinh(m(k))/cosh(m) is algebraically false.
  2. [Section 3, proof of the parity lemma] The notation 'Gamma(partial_1)' and 'Gamma(partial_n)' should be Gamma(x_1) and Gamma(x_2), and the comparison of Gaussian densities should be at (a_1,a_2) and (a_1,-a_2), not (a_1,-a_1).
  3. [Sections 4.4 and 5] Several consequences, including the explicit bijection via 'peeling' and the proof of Lupu's intensity doubling conjecture, are deferred to papers listed as in preparation ([46], [68], [69], [13]). The reader would benefit from a sentence making explicit which statements are conditional on those forthcoming works.
  4. [Throughout] There are several typographical slips: 'loose their full independence' should be 'lose their full independence'; in Section 3, 'm(k)' is used before its definition in the same sentence; and the proof of Lemma 6 refers to 'x' and 'y' without restating their role.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the switching theorem is derived from standard couplings plus re-proved parity lemmas; self-citations are not load-bearing.

full rationale

The central claim, Theorem 2, is not assumed as an input anywhere in the paper. The Section 2 proof derives it from the standard loop-soup/GFF coupling, the rewiring property of [66], and a detailed discrete two-state chain calculation. The parity lemma (Lemma 6) is explicitly not taken on faith: the paper states that it is 'almost exactly Proposition 7 in [66]' but then says 'we will provide some more details about this proof' and proves Lemma 8 from the loop-soup formula (2), with a controlled K-to-infinity limit. Section 3 then gives an independent Laplace-transform/Dynkin-isomorphism route in which the four independent Poisson inputs are identified from the Laplace transforms of occupation fields and the even/odd conditioning is obtained by symmetrizing over Phi1+Phi2 versus Phi1-Phi2. Self-citations to [66], [44], and [35] are used for background, rewiring properties, or related prior results, but none of those cited results contains the switching theorem, and the paper does not rely on a self-citation chain to force the conclusion. The possible factor-of-two normalization point in Section 3 is a mathematical-correctness concern about the covariance convention, not a circularity of the derivation; even if that route needed repair, the Section 2 proof supplies an independent derivation. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is smuggled in via author citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted: a and b are conditioning values and the Poisson means are determined by excursion measures. The load-bearing background consists of the GFF/loop-soup correspondence and prior parity and rewiring results, which are cited rather than re-derived. No new physical entities are postulated; proof devices such as the ghost edge are auxiliary graph gadgets with no separate falsifiable content.

assumptions (6)
  • domain assumption GFF/loop-soup correspondence: the occupation field of a critical Brownian loop-soup has the law of Gamma squared, and loop clusters equal sign-clusters.
    Invoked throughout as the bridge between loop-soup and GFF; proven in Le Jan and Lupu works cited in the Introduction.
  • domain assumption Rewiring/spatial Markov property for unoriented loop-soups, Proposition 7 of [66].
    Used as the base for Lemma 6 and Lemma 13; the present paper fills in the proof sketch from [66].
  • standard math Dynkin's isomorphism and the Poisson decomposition of excursions produced by conditioned GFF squares.
    Section 3's Laplace-transform proof and the excursion interpretations rely on it; standard in the cited literature.
  • standard math Reflection principle for cable-graph GFF: sign clusters can be resampled independently.
    Used to compare probabilities of same-sign versus different-sign at x and y in the switching proof.
  • domain assumption Convergence of cable-graph excursion measures from x to y_n to an excursion measure from x to infinity under a stated hypothesis.
    Hypothesis of Proposition 23 and used for Theorem 3; for Zd this is classical, for general graphs it is an explicit assumption.
  • domain assumption Effective resistance reduction: the cable graph can be replaced by a single edge for excursion and parity computations.
    Remark 11 and Section 4 rely on equivalence ideas from [45] to move from one-edge to general graph switching.

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Cite this review

Pith. "Pith review of A switching identity for cable-graph loop soups and Gaussian free fields." pith.science (2026). https://pith.science/paper/JMAOOGTX

@misc{pith2026250206754,
  author       = {Pith},
  title        = {Pith review of: A switching identity for cable-graph loop soups and Gaussian free fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMAOOGTX}},
  note         = {Machine review of arXiv:2502.06754}
}
read the original abstract

We derive a "switching identity" that can be stated for critical Brownian loop-soups or for the Gaussian free field on a cable graph: It basically says that at the level of cluster configurations and at the more general level of the occupation time fields, conditioning two points on the cable-graph to belong to the same cluster of Brownian loops (or equivalently to the same sign-cluster of the GFF) amounts to adding a random odd number of independent Brownian excursions between these points to an otherwise unconditioned configuration. This explicit simple description of the conditional law of the clusters when a connection occurs has various direct consequences, in particular about the large scale behaviour of these sign-clusters on infinite graphs.

Figures

Figures reproduced from arXiv: 2502.06754 by the authors.

Figure 1
Figure 1. The two sketched constructions (conditioning on connecting x and y vs. just adding one excursion between x and y) give the same measure on clusters. 1 arXiv:2502.06754v4 [math.PR] 3 Jul 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

71 extracted references · 66 canonical work pages · cited by 2 Pith papers

  1. [35]

    Parity questions in critical planar Brownian loop-soups (or ”where did the free planar bosons go?”)

    Matthis Lehmkuehler, Wei Qian, and Wendelin Werner. Parity questions in critical planar Brownian loop-soups (or ”where did the free planar bosons go?”). arXiv 2403.07830, 2024

    arXiv 2024

  2. [1]

    Cluster explorations of the loop soup on a metric graph related to the Gaussian free field

    Elie A ¨ ıd´ ekon. Cluster explorations of the loop soup on a metric graph related to the Gaussian free field. arXiv:2009.05120

    work page Pith review arXiv 2009

  3. [66]

    On the spatial Markov property of soups of unoriented and oriented loops

    Wendelin Werner. On the spatial Markov property of soups of unoriented and oriented loops. In S´ eminaire de Probabilit´ es XL VIII, L.N. in Math. 2168, 481-503. Springer, 2016

    work page 2016

  4. [44]

    Titus Lupu and Wendelin Werner: A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field. Electr. Comm. Probab. 21, paper 13, 2016

    work page 2016

  5. [2]

    The stochastic Jacobi flow

    Elie A ¨ ıd´ ekon, Yueyun Hu, Zhan Shi. The stochastic Jacobi flow. arXiv:2306.12716

    work page Pith review arXiv

  6. [3]

    Geometric Analysis of ϕ4 Fields and Ising Models (Parts 1 & 2)

    Michael Aizenman. Geometric Analysis of ϕ4 Fields and Ising Models (Parts 1 & 2). Comm. Math. Phys. 86, 1-48, 1982

    work page 1982

  7. [4]

    On the number of incipient spanning clusters

    Michael Aizenman. On the number of incipient spanning clusters. Nuclear Phys. B 485, 551-582, 1997

    work page 1997

  8. [5]

    David J. Aldous. Brownian excursion conditioned on its local time. Electronic Comm. Probab. 3, 79-90, 1998

    work page 1998

Show all 71 references
  1. [6]

    Excursion decomposition of the 2d continuum GFF

    Juhan Aru, Titus Lupu, and Avelio Sep´ ulveda. Excursion decomposition of the 2d continuum GFF. arXiv 2304.03150, 2023

    2023 arXiv

  2. [7]

    Spin systems from loop soups

    Tim van de Brug, Federico Camia, Marcin Lis. Spin systems from loop soups. Electr. J. Probab. 23, paper no. 81, 1–17, 2018

    2018

  3. [8]

    The random walk representation of classical spin systems and correlation inequalities

    David Brydges, J¨ urg Fr¨ ohlich, and Tom Spencer. The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys. 83, 123-150, 1982

    1982

  4. [9]

    One-arm exponent of critical level-set for metric graph Gaussian free field in high dimensions

    Zhenhao Cai and Jian Ding. One-arm exponent of critical level-set for metric graph Gaussian free field in high dimensions. arXiv:2307.04434

    arXiv

  5. [10]

    One-arm Probabilities for metric graph Gaussian free fields below and at the Critical dimension

    Zhenhao Cai and Jian Ding. One-arm Probabilities for metric graph Gaussian free fields below and at the Critical dimension. arXiv 2406.02397, 2024. 33

    2024 arXiv

  6. [11]

    Quasi-multiplicativity and regularity for critical metric graph Gaussian free fields, arXiv 2412.05706, 2024

    Zhenhao Cai and Jian Ding. Quasi-multiplicativity and regularity for critical metric graph Gaussian free fields, arXiv 2412.05706, 2024

    2024 arXiv

  7. [12]

    Incipient infinite clusters and self-similarity for metric graph Gaussian free fields and loop soups, arXiv 2412.05709, 2024

    Zhenhao Cai and Jian Ding. Incipient infinite clusters and self-similarity for metric graph Gaussian free fields and loop soups, arXiv 2412.05709, 2024

    2024 arXiv

  8. [13]

    On loops in critical high-dimensional percolation

    Amelia Carpenter and Wendelin Werner. On loops in critical high-dimensional percolation. arXiV, 2025

    2025

  9. [14]

    Percolation for level-sets of Gaussian free fields on metric graphs

    Jian Ding and Mateo Wirth. Percolation for level-sets of Gaussian free fields on metric graphs. Ann. Probab. 48, 1411–1435, 2020

    2020

  10. [15]

    Cluster capacity functionals and isomorphism theorems for Gaussian free fields, Probab

    Alex Drewitz, Alexis Pr´ evost, and Pierre-Fran¸ cois Rodriguez. Cluster capacity functionals and isomorphism theorems for Gaussian free fields, Probab. Theory rel. Fields 183, 255-313, 2022

    2022

  11. [16]

    Critical exponents for a percolation model on transient graphs

    Alex Drewitz, Alexis Pr´ evost, and Pierre-Fran¸ cois Rodriguez. Critical exponents for a percolation model on transient graphs. Inventiones Math. 232, 229–299, 2023

    2023

  12. [17]

    Critical one-arm probability for the metric Gauss- ian free field in low dimensions

    Alex Drewitz, Alexis Pr´ evost, and Pierre-Fran¸ cois Rodriguez. Critical one-arm probability for the metric Gauss- ian free field in low dimensions. arXiv:2405.17417, 2024

    2024 arXiv

  13. [18]

    Cluster volumes for the Gaussian free field on metric graphs

    Alex Drewitz, Alexis Pr´ evost, and Pierre-Fran¸ cois Rodriguez. Cluster volumes for the Gaussian free field on metric graphs. arXiv2412.06772, 2024

    2024 arXiv

  14. [19]

    SLE and the Free Field: Partition functions and couplings

    Julien Dub´ edat. SLE and the Free Field: Partition functions and couplings. J. Amer. Math. Soc. 22, 995–1054, 2009

    2009

  15. [20]

    Hugo Duminil-Copin, 100 years of the (critical) Ising model on the hypercubic lattice, arXiv:2208.00864, 2022

    2022 arXiv

  16. [21]

    Eugene B. Dynkin. Markov processes as a tool in Field Theory. J. Funct. Anal. 50, 167-187, 1983

    1983

  17. [22]

    Eugene B. Dynkin. Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55, 344-376, 1984

    1984

  18. [23]

    Critical level set percolation for the GFF in d >6: comparison principles and some consequences

    Shirshendu Ganguly and Kaihao Jing. Critical level set percolation for the GFF in d >6: comparison principles and some consequences. arXiv:2412.17768

    arXiv

  19. [24]

    The ant on loops: Alexander-Orbach conjecture for the critical level set of the Gaussian free field

    Shirshendu Ganguly and Kyeongsik Nam. The ant on loops: Alexander-Orbach conjecture for the critical level set of the Gaussian free field. arXiv:2403.02318, 2024

    2024 arXiv

  20. [25]

    Griffiths, Charles A

    Robert B. Griffiths, Charles A. Hurst and Seymour Sherman. Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11, 790–795, 1970

    1970

  21. [26]

    Random even graphs

    Geoffrey Grimmett and Svante Jansson. Random even graphs. Electr. J. Combin. 16, # R46, 2009

    2009

  22. [27]

    Conformally invariant fields out of Brownian loop soups

    Antoine Jego, Titus Lupu and Wei Qian. Conformally invariant fields out of Brownian loop soups. arXiv:2307.10740

    arXiv

  23. [28]

    Covariant Symanzik identities

    Adrien Kassel and Thierry L´ evy. Covariant Symanzik identities. Probability and Mathematical Physics, 2, 419–475, 2021

    2021

  24. [29]

    The incipient infinite cluster in two-dimensional percolation

    Harry Kesten. The incipient infinite cluster in two-dimensional percolation. Probab. Th. Rel. Fields 73, 369–394, 1986

    1986

  25. [30]

    The Alexander-Orbach conjecture holds in high dimensions

    Gady Kozma and Asaf Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178, 635–654, 2009

    2009

  26. [31]

    Gregory F. Lawler. Topics in loop measures and the loop-erased walk. Probab. Surveys 15, 28-101, 2018

    2018

  27. [32]

    Lawler, Oded Schramm, and Wendelin Werner

    Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: The chordal case. J. Amer. Math. Soc. 16, 917-955, 2003

    2003

  28. [33]

    Lawler and Jos´ e A

    Gregory F. Lawler and Jos´ e A. Trujillo Ferreras. Random Walk Loop Soup. Trans. Amer. Math. Soc. 359, 767-787, 2007

    2007

  29. [34]

    Lawler and Wendelin Werner

    Gregory F. Lawler and Wendelin Werner. The Brownian loop soup. Probab. Th. rel. Fields 128, 565-588, 2004

    2004

  30. [36]

    Markov loops and renormalization

    Yves Le Jan. Markov loops and renormalization. Ann. Probab. 38, 1280-1319, 2010

    2010

  31. [37]

    Markov paths, loops and fields , L.N

    Yves Le Jan. Markov paths, loops and fields , L.N. in Math. 2026, Springer, Heidelberg, 2011. Lectures from the 38th Probability Summer School held in Saint-Flour, 2008

    2026

  32. [38]

    Random Walks and Physical Fields, Springer, 2024

    Yves Le Jan. Random Walks and Physical Fields, Springer, 2024

    2024

  33. [39]

    From loop clusters and random interlacements to the free field

    Titus Lupu. From loop clusters and random interlacements to the free field. Ann. Probab. 44, 2117-2146, 2016

    2016

  34. [40]

    Convergence of the two-dimensional random walk loop-soup clusters to CLE

    Titus Lupu. Convergence of the two-dimensional random walk loop-soup clusters to CLE. J. Eur. Math. Soc. 21, 1201-1227, 2019

    2019

  35. [41]

    Titus Lupu, An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields, arXiv 2209.07901, 2023

    2023 arXiv

  36. [42]

    Inverting the coupling of the signed Gaussian free field with a loop-soup

    Titus Lupu, Christophe Sabot, and Pierre Tarr` es. Inverting the coupling of the signed Gaussian free field with a loop-soup. Electron. J. Probab. 24, paper 70, 2019

    2019

  37. [43]

    Fine mesh limit of the VRJP in dimension one and Bass–Burdzy flow

    Titus Lupu, Christophe Sabot, and Pierre Tarr` es. Fine mesh limit of the VRJP in dimension one and Bass–Burdzy flow. Probab. Theory rel. fields 177, 55–90, 2020. 34

    2020

  38. [45]

    The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

    Titus Lupu and Wendelin Werner. The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph. Probab. Theory rel. Fields 171, 775-818, 2018

    2018

  39. [46]

    In preparation

    Titus Lupu and Wendelin Werner. In preparation

  40. [47]

    Marcus and Jay Rosen

    Michael B. Marcus and Jay Rosen. Markov processes, Gaussian processes, and local times. Cambridge University Press, 2006

    2006

  41. [48]

    private communication and talks, 2010

    Jason Miller and Scott Sheffield. private communication and talks, 2010

    2010

  42. [49]

    CLE percolations

    Jason Miller, Scott Sheffield, and Wendelin Werner. CLE percolations. Forum Math. Pi, 5:e4, 102 pages, 2017

    2017

  43. [50]

    Connection probabilities for conformal loop ensembles

    Jason Miller and Wendelin Werner. Connection probabilities for conformal loop ensembles. Comm. Math. Phys. 362, 415-453, 2018

    2018

  44. [51]

    A decomposition of Bessel Bridges

    Jim Pitman and Marc Yor. A decomposition of Bessel Bridges. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 425–457, 1982

    1982

  45. [52]

    Percolation for the Gaussian free field on the cable system: counterexamples

    Alexis Pr´ evost. Percolation for the Gaussian free field on the cable system: counterexamples. Electron. J. Probab. 28, paper 62 (2023)

    2023

  46. [53]

    Conditioning a Brownian loop-soup cluster on a portion of its boundary

    Wei Qian. Conditioning a Brownian loop-soup cluster on a portion of its boundary. Ann. de Inst. H. Poincar´ e (B) 55, 314-240, 2019

    2019

  47. [54]

    The law of a point process of Brownian excursions in a domain is determined by the law of its trace

    Wei Qian and Wendelin Werner. The law of a point process of Brownian excursions in a domain is determined by the law of its trace. Electron. J. Probab. 23, Paper 128, 2018

    2018

  48. [55]

    Decomposition of Brownian loop-soup clusters

    Wei Qian and Wendelin Werner. Decomposition of Brownian loop-soup clusters. J. Eur. Math. Soc. 21, 3225-3253, 2019

    2019

  49. [56]

    L.C.G. Rogers. Williams’ characterization of the Brownian excursion law: proof and applications. In S´ eminaire de Probabilit´ es XV, Lecture Notes in Math. 850, 227-250. Springer, 1981

    1981

  50. [57]

    Inverting Ray-Knight identity

    Christophe Sabot and Pierre Tarr` es. Inverting Ray-Knight identity. Probab. Theory Related Fields 165, 559-580, 2015

    2015

  51. [58]

    A contour line of the continuum Gaussian free field

    Oded Schramm and Scott Sheffield. A contour line of the continuum Gaussian free field. Probab. Theory rel. Fields 157, 47-80, 2013

    2013

  52. [59]

    Conformal loop ensembles: the Markovian characterization and the loop- soup construction

    Scott Sheffield and Wendelin Werner. Conformal loop ensembles: the Markovian characterization and the loop- soup construction. Ann. of Math. (2) 176, 1827-1917, 2012

    1917

  53. [60]

    Topics in occupation times and Gaussian free field

    Alain-Sol Sznitman. Topics in occupation times and Gaussian free field. Z¨ urich lectures in advanced mathematics. Europ. Math. Soc., 2012

    2012

  54. [61]

    On scaling limits and Brownian interlacements

    Alain-Sol Sznitman. On scaling limits and Brownian interlacements. Bull. Braz. Math. Soc., 44, 55-592, 2013

    2013

  55. [62]

    Euclidean quantum field theory

    Kurt Symanzik. Euclidean quantum field theory. In: Local quantum theory. (ed. Jost) , Academic Press, 1969

    1969

  56. [63]

    The true self-repelling motion

    B´ alint T´ oth and Wendelin Werner. The true self-repelling motion. Probab Theory Relat Fields 111, 375–452, 1998

    1998

  57. [64]

    A stochastic flow arising in the study of local times

    Jon Warren. A stochastic flow arising in the study of local times. Probab. Theory rel. Fields, 133:559–572, 2005

    2005

  58. [65]

    The Brownian Burglar: Conditioning Brownian motion by its local time process

    Jon Warren and Marc Yor. The Brownian Burglar: Conditioning Brownian motion by its local time process. In S´ eminaire de Probabilit´ es XXXII, L.N. in Math. 1686, 328–342, Springer, 1998

    1998

  59. [67]

    On clusters of Brownian loops in d dimensions

    Wendelin Werner. On clusters of Brownian loops in d dimensions. In In and out of equilibrium 3. Celebrating Vladas Sidoravicius, Progr. Probab. 77, 797–817, Birkh¨ auser/Springer, 2021

    2021

  60. [68]

    Loop-soup percolation in high dimensions

    Wendelin Werner. Loop-soup percolation in high dimensions. In preparation

  61. [69]

    Loop-soup percolation in three (and four) dimensions

    Wendelin Werner. Loop-soup percolation in three (and four) dimensions. In preparation

  62. [70]

    Lecture notes on the Gaussian free field , volume 28 of Cours Sp´ ecialis´ es

    Wendelin Werner and Ellen Powell. Lecture notes on the Gaussian free field , volume 28 of Cours Sp´ ecialis´ es. Soci´ et´ e Math´ ematique de France, 2021

    2021

  63. [71]

    Some Aspects of Brownian Motion Part I : Some Special Functionals

    Marc Yor. Some Aspects of Brownian Motion Part I : Some Special Functionals . Lectures in Mathematics, ETH Z¨ urich, Birkh¨ auser, 1992. University of Cambridge 35

    1992

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