Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

Alberto Gandolfi; Federico Camia; Giovanni Peccati; Tulasi Ram Reddy

Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Brownian loop-soup layering fields converge to a tilted imaginary Gaussian multiplicative chaos.

desk verdict Careful, substantial proof that Brownian loop soup layering fields converge to tilted imaginary GMC; the main caveat is a standard but load-bearing small-loop constant imported from earlier work. read the letter →

arxiv 1908.05881 v2 pith:WPPZE2BF submitted 2019-08-16 math.PR math-phmath.MP

Federico Camia , Alberto Gandolfi , Giovanni Peccati , Tulasi Ram Reddy This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60G6060D0560F0560H0760J65

keywords BrownianloopsouplayeringfieldsimaginaryGaussianmultiplicativechaosWiener-ItôexpansionconformalcovariancenegativeSobolevspacesmassivediskmodel

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 studies vertex-like fields built from the Brownian loop soup: at each point of a planar domain one takes the exponential of an imaginary coupling constant times the signed number of loops that surround the point, each loop carrying an independent random sign. Since infinitely many small loops surround every point, the fields are defined with a diameter cutoff and renormalized by a power of the cutoff; the paper proves that these renormalized fields exist as random generalized functions. It then sends the loop-soup intensity to infinity while the coupling shrinks, with $\lambda\beta^2\to\xi^2<5$, and proves that the Poissonian fields converge to a Gaussian layering field, in finite-dimensional distributions for general domains and in the negative Sobolev space $H^{-\alpha}$ for bounded $C^1$ domains. The limit is a deterministically tilted imaginary Gaussian multiplicative chaos whose covariance kernel is the Brownian loop measure of loops disconnecting two points from the boundary. This gives a rigorous bridge between a Poissonian loop-soup model and the continuum theory of log-correlated Gaussian fields, and supplies a new non-Gaussian construction of imaginary Gaussian multiplicative chaos.

What carries the argument

The main tool is an explicit Wiener-Itô chaos expansion of the action of the fields on test functions, together with convergence of compensated Poisson chaos terms to Gaussian chaos terms. For the Poisson layering field the $q$-th chaos kernel is an integral of the one-point function against $(e^{i\beta h_z}-1)^{\otimes q}$; in the limit $\lambda\to\infty$, $\beta\to 0$ with $\lambda\beta^2\to\xi^2$, the identity $\lambda^{q/2}(e^{i\beta h_z}-1)^{\otimes q}\to (i\xi h_z)^{\otimes q}$ transfers every chaos term to the Gaussian chaos term, and uniform summability is controlled by the bound $\alpha^*_D(z,w)\le (1/5)\log(2/|z-w|)$. The exact small-loop divergence rates $\alpha^{\rm loop}_{\delta,R}(z)=(1/5)\log(R/\delta)$ and $\alpha^{\rm disk}_{\delta,R}(z)=\pi\log(R/\delta)$, quoted from [15, Lemma A.1], fix the renormalization exponents $\Delta^{\rm loop}_{\lambda,\beta}=(\lambda/10)(1-\cos\beta)$ and $\Delta^{\rm disk}_{\lambda,\beta}=(\lambda\pi/2)(1-\cos\beta)$, the coefficient $1/5$ in the covariance singularity, and the convergence threshold $\xi^2<5$.

What would settle it

Count the expected number of Brownian loop-soup loops of diameter between $\delta$ and $R$ that surround a fixed point, for example by simulation or by a rigorous Brownian-bridge estimate; if the growth is not $\alpha^{\rm loop}_{\delta,R}(z)=(1/5)\log(R/\delta)+o(1)$ as $\delta\downarrow 0$, then the renormalization exponents, the covariance kernel, and the convergence claim change.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 6.2 and Theorem 1.1: for the loop and massive loop models, as $\lambda\to\infty$ and $\beta\to 0$ with $\lambda\beta^2\to\xi^2<5$, the renormalized Poisson layering fields $V^*_{\lambda,\beta}$ converge to the Gaussian layering field $W^*_\xi$. For bounded simply connected domains with $C^1$ boundary the convergence holds in distribution in $H^{-\alpha}$ for every $\alpha>3/2$; for arbitrary domains conformally equivalent to the disk it holds in the sense of finite-dimensional distributions. The limit satisfies $dW^*_{\xi,D}/dM^*_{\xi,D}(z)=e^{-\xi^2\Theta^*_D(z)/2}$, where $M^*_{\xi,D}$ is an imaginary Gaussian multiplicative chaos with parameter $\xi$ and covariance kernel $K^*_D(z,w)=\mu^*(\gamma:\gamma\subset D,\ \gamma\text{ disconnects }z,w\text{ from }\partial D)$, and $\Theta^*_D$ is an explicit deterministic tilt built from the loop measure and the distance to the boundary. The covariance kernel is not the Green's function of the Laplacian, so the Gaussian limit is not a free field; for the disk model the conclusion holds with $\xi^2<1/\pi$ and conformal dimension $\pi\xi^2/4$.

Load-bearing premise

The argument rests on the exact logarithmic rate at which the Brownian loop measure diverges for small loops—coefficient $1/5$ for loops and $\pi$ for disks, quoted from [15, Lemma A.1]; if that rate were different, the renormalization exponents, the covariance singularity, the conformal dimension $\xi^2/20$, and the convergence threshold $\xi^2<5$ would all change.

Editorial extensions

If this is right

The renormalized layering field exists as a random generalized function in $H^{-\alpha}$, $\alpha>3/2$, so the correlation functions previously derived in [15] are realized by an actual limiting field and not only at the level of moments.
In the high-intensity, small-coupling regime the Poissonian loop soup becomes Gaussian: all randomness of the limit is carried by an imaginary Gaussian multiplicative chaos with Brownian-loop covariance, and the tilt factor is deterministic.
The limiting loop and massive fields are conformally covariant with scaling dimension $\xi^2/20$ (disk: $\pi\xi^2/4$), so they transform like vertex operators of that dimension under conformal changes of domain.
The massive loop soup converges by the same mechanism, with the same dimension as the massless case and only the tilt modified by the killing factor.
Because the limiting covariance is $K^*_D$, not the Green's function, the Gaussian limit is a log-correlated field of a new explicit type rather than the free-field limit suggested by earlier heuristics.

Reading between the lines

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

An editorial extension: the same chaos-expansion mechanism should apply to winding fields and other exponential functionals of loop soups, potentially producing Gaussian limits in parameter regimes where the unrenormalized field is non-Gaussian.
The threshold $\xi^2<5$ is left open as possibly an artifact of the method; one test of sharpness is to monitor the high-order chaos norms near $\xi^2=5$, where the bound involving $\alpha^*_D(z,w)\simeq (1/5)\log(2/|z-w|)$ stops giving uniform summability.
Because the covariance kernel in the unit disk has an explicit hypergeometric formula (quoted from [28] in the paper), a numerical check of the two-point function of the finite-$\delta$ fields against the predicted iGMC covariance would provide a quantitative test of the convergence.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things you should know. First, this is the real thing: Camia et al. prove that renormalized Poisson layering fields built from the Brownian loop soup converge, in the high-intensity/small-coupling limit, to an explicitly identified tilted imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Second, the central machinery—Wiener-Itô chaos expansions for the Poisson and Gaussian fields, then matching kernels term by term—is new and works. This is the first rigorous construction of imaginary GMC from a non-Gaussian Poissonian model, and it settles the informal free-field suggestion in the disk model literature.

The paper is honest about its limits. The convergence is f.d.d.-only for general domains; Sobolev-space convergence is proved for bounded C¹ domains. And the existence of the Gaussian layering field is established for ξ²<10 but convergence only for ξ²<5; they flag that gap explicitly. I checked the main steps: the renormalization constants follow from the small-loop divergence rates, two-point bounds satisfy the abstract Theorem 3.3, and the chaos expansion convergence is checked with dominated convergence and tail control. The appendix proofs are substantial, not filler.

Where are the soft spots? The one load-bearing input not re-derived here is the exact small-loop divergence rate α^loop_{δ,R}(z) = (1/5) log(R/δ) (and π log(R/δ) for disks), quoted from [15] and [25]. Everything quantitative hangs off that constant: the conformal dimension ξ²/20, the covariance singularity coefficient, and the threshold ξ²<5. If that constant were off, the results would shift. But it is a standard, externally supported fact about the Brownian loop measure, and the paper uses it consistently; I do not see a circularity or an internal inconsistency. The other caveat is reliance on the imaginary GMC existence theorem of Junnila–Saksman–Webb, which is also standard. These are residual risks rather than flaws.

The citation pattern is fair. They cite their own prior work for the one-point function and loop soup covariance, but those are standard Poisson computations and are not the target result. The new theorems are genuinely new.

Who is this for? People working in conformally invariant stochastic fields, loop soups, and GMC. It is a serious 67-page proof paper; a referee needs time. I would send it to a good referee. My own verdict is accept with no major revisions needed—maybe ask them to add a remark explicitly saying where the 1/5 constant comes from and what would change if it were different, but that is a courtesy, not a requirement.

Referee Report

0 major / 5 minor

Summary. The paper studies renormalized vertex-like layering fields built from the Brownian loop soup, the massive Brownian loop soup, and a scale-invariant disk model. Each loop receives an independent random sign, and the field at a point is the exponential of an imaginary constant times the signed number of loops winding around that point. After an ultraviolet cutoff δ, the authors prove in Theorem 4.5 that the renormalized fields converge as δ→0 in negative Sobolev spaces H^{-α}, α>3/2, to Poisson layering fields V^*_{λ,β}; a parallel construction in Theorem 4.9 gives Gaussian layering fields W^*_ξ. The main result, Theorem 6.2 (with Theorem 1.1), shows that as λ→∞ and β→0 with λβ²→ξ²<5, the Poisson layering fields converge in finite-dimensional distributions to W^*_ξ, which is expressed as a tilted imaginary Gaussian multiplicative chaos with covariance kernel K^*_D(z,w)=μ^*(γ: γ in D, disconnects z,w from ∂D) and explicit density exp(-ξ²Θ^*_D(z)/2). For bounded C1 domains the convergence is upgraded to distributional convergence in H^{-α} via tightness and a uniqueness lemma. The proofs combine a general existence theorem for exponentially integrated Poisson/Gaussian fields, explicit one- and two-point estimates for the loop measures, and Wiener-Itô chaos expansions whose kernels are shown to converge term by term with uniform tail control.

Significance. This is a substantial contribution if its claims hold. It gives the first rigorous construction, to my knowledge, of the limiting layering fields from the Brownian loop soup as imaginary Gaussian multiplicative chaos, and it provides a new non-Gaussian route to imaginary GMC. The conformal covariance statement in Theorem 1.3, with conformal dimension ξ²/20 for the loop and massive loop cases and πξ²/4 for the disk case, is explicit and falsifiable. The proof strategy is a genuine methodological contribution: the Wiener-Itô chaos expansion reduces the asymptotic analysis to one-point functions and two-point kernel estimates, and the term-by-term verification of the convergence conditions is unusually detailed. The paper also credits and builds on prior work in a transparent way, and the residual risk identified in the stress-test analysis — the exact small-loop divergence constants in (4.13)–(4.14), quoted from [15] and [25] — is a standard, externally documented input rather than an internal inconsistency. I found no circularity and no free parameters in the derivation.

minor comments (5)

[4.1, Eqs. (4.13)–(4.14)] Because the constants 1/5 and π in (4.13)–(4.14) determine the renormalization exponents, the covariance singularity, and the thresholds ξ²<5 and ξ²<1/π, I suggest adding one sentence making explicit that (4.13) is exactly Lemma A.1 of [15] with the normalization of μ^loop used there, and that (4.14) follows from the computation in [25, Section 3.1]. This is implicit in the current citation, but stating it verbatim would remove all ambiguity about the normalization of the Brownian loop measure.
[5.2, Eq. (5.43)] In the Gaussian massive case, the displayed constant C_* appears to be copied from the Poisson bound (5.19). Since the relevant Gaussian exponent is -ξ²/2 times α^m_{δ,D}(z), the massive factor should involve exp(ξ² times the corresponding limiting massive-loop mass) rather than exp(2λ(1-cosβ) times that mass). As written, the bound is still finite for the fixed parameters used in Theorem 5.3, but the formula is formally a leftover of the Poisson calculation and should be corrected.
[5.2, proof of Theorem 5.3] In the paragraph immediately after (5.36), the notation V^*_ξ(ϕ) is used where W^*_ξ(ϕ) is clearly intended, and the sentence 'admit a chaos expansions' should read 'admit chaos expansions.' These are typographical issues only, but they appear in the statement of a central technical result.
[1.3, Theorem 1.3, displays (1.10)–(1.12)] In the first integral of each displayed identity, the test function is written as φ(w) while the integration variable is dz; the argument should be z, so that the first integrand reads W^*_{ξ,D}(z)φ(z)dz. As printed, the notation is not typographically consistent with the following line, where the change of variables to w is made.
[A.5, Eqs. (A.97)–(A.98)] In the Gaussian part of the proof of Theorem A.6, the exponent Δ^*_{λ,β} appears where Δ^*_ξ is meant; no Gaussian field with parameters λ and β has been defined in that section. The displayed formulas are otherwise clear, but this notational slip should be fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Gaussian limit is constructed from the Brownian loop measure, and the convergence proof verifies a CLT via chaos expansions; self-citations are to standard parameter-free loop-soup computations, not to the target result.

full rationale

The paper's central claim is that Poisson layering fields V*_{λ,β} converge to Gaussian layering fields W*_ξ, where W*_ξ is a tilted imaginary Gaussian multiplicative chaos with covariance kernel K*_D(z,w)=μ*(γ in D disconnects z,w from ∂D). This target is not used as an input: W*_ξ is constructed independently (Section 4.5) from a Gaussian measure with control given by the Brownian loop measure, and the covariance kernel is computed from the loop measure itself, not fitted to the Poisson field. The proof of Theorem 6.2 proceeds by deriving explicit Wiener-Itô chaos expansions for both fields and verifying the CLT-type conditions (6.5)–(6.8), i.e., convergence of one-point functions, square-integrability of Gaussian chaos kernels, kernel-by-kernel convergence, and asymptotic vanishing of chaos tails. No parameter is fitted to a subset of the data and then renamed a prediction: the renormalization exponents Δ^loop_{λ,β}=λ/10(1−cosβ) and Δ^disk_{λ,β}=λπ/2(1−cosβ) come from the small-loop divergence rates quoted in (4.13)–(4.14), and the convergence threshold ξ²<5 follows from the explicit bound α^loop_D(z,t)≤(1/5)log(2/|z−t|) used in (6.20), (6.21), (6.33). The cited results from the authors' prior work [15] and [13] are parameter-free, standard loop-soup computations that do not include the target convergence statement; they are externally falsifiable and have independent derivations in the Brownian loop soup literature. No self-citation chain is used to forbid alternatives or to force the choice of the limiting field. The residual risk that the constant 1/5 in (4.13) is quoted rather than re-derived is a correctness/verification concern, not a circularity: the derivation is self-contained once that standard Brownian loop measure fact is granted.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; λ, β and ξ are model parameters, and the constants 1/5 and π are fixed properties of the measures. The paper introduces no new physical entities; the layering fields are constructed from the existing Brownian loop soup and Gaussian measures. The main external inputs are standard theorems from [15], [45], [30] and [13], none of which contains the convergence result.

assumptions (5)

standard math Brownian loop measure is conformally invariant, restriction-invariant, and has small-loop divergence α^{loop}_{δ,R}(z)=(1/5)log(R/δ); the disk measure has α^{disk}_{δ,R}(z)=π log(R/δ).
Taken from [15, Lemma A.1] and Werner's loop measure theory; used to fix renormalization exponents and the covariance singularity.
standard math Brownian loop soup is thin: lim_{R→∞} μ(γ: γ∩D≠∅, diam(γ)≥R)=0.
From Nacu-Werner [45]; used in Theorem 4.2 and Lemma 4.4 to make two-point limits finite.
standard math Imaginary GMC exists for log-correlated Gaussian fields with covariance log^+ 1/|z-w| plus a bounded continuous function, under the standard approximation conditions of Junnila-Saksman-Webb.
Black-box use of [30, Theorem 1.1] to identify the limiting field as imaginary GMC.
standard math The one-point function of a Poisson layering field with cutoff δ is e^{-λ α^*_{δ,D}(z)(1-cosβ)}.
Kintchine formula; recorded as Eq. (4.2) of [15] and used in Lemma A.3 and Section 5.
domain assumption For bounded C1 domains, eigenfunction sup-norm bounds and Weyl asymptotics make the H^{-α} Sobolev argument work; for general conformal domains only f.d.d. convergence is asserted.
Assumption on the domain in Theorems 1.1 and 6.2; the C1 boundary is needed for tightness in H^{-α}.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos." pith.science (2026). https://pith.science/paper/WPPZE2BF

@misc{pith2026190805881,
  author       = {Pith},
  title        = {Pith review of: Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPPZE2BF}},
  note         = {Machine review of arXiv:1908.05881}
}

read the original abstract

We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of the loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener-It\^{o} chaos expansion of linear functionals of vertex-like fields.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 63 canonical work pages

[15]

Camia, A

F. Camia, A. Gandolﬁ, and M. Kleban. Conformal correlation functions in the Brownian loop soup. Nuclear Physics B , 902:483–507, 2016

work page 2016
[25]

Freivogel and M

B. Freivogel and M. Kleban. A conformal ﬁeld theory for eternal inﬂation ? J. High Energy Phys., (12):019, 31, 2009

work page 2009
[1]

Aizenman

M. Aizenman. Geometric Analysis of φ4 Fields and Ising Models. Parts I and II, Comm. Math. Phys. 86: 1–48, 1982

work page 1982
[2]

Astala, P

K. Astala, P. Jones, A. Kupiainen, and E. Saksman. Random conformal weldings. Acta Math. 207 , no. 2: 203–254, 2011

work page 2011
[3]

Belavin, A.M

A.A. Belavin, A.M. Polyakov, and A.B. Zamolodchikov. Inﬁnite conformal symmetry in two-dimensional quantum ﬁeld theory. Nuclear Phys. B , 241(2):333–380, 1984

work page 1984
[4]

Belavin, A.M

A.A. Belavin, A.M. Polyakov, and A.B. Zamolodchikov. Inﬁnite conformal symmetry of critical ﬂuctuations in two dimensions. J. Statist. Phys. , 34(5-6):763–774, 1984

work page 1984
[5]

Berestycki

N. Berestycki. An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab., 22:Paper No. 27, 12, 2017

work page 2017
[6]

Berestycki, C

N. Berestycki, C. Webb, and M. D. Wong. Random Hermitian matrices and Gaussian multiplicative chaos. Probab. Theory Related Fields, 172(1-2):103–189, 2018

work page 2018

Show all 66 references

[7]

L. Bers, F. John, and M. Schechter. Partial diﬀerential equations. Lectures in Applied Mathematics 3A, American Mathematical Society, 1964

1964
[8]

Borodin and P

A. Borodin and P. Ferrari Anisotropic growth of random surfaces in 2+1 dimensions. Comm.Math. Phys. 325, no. 2: 603—684, 2014

2014
[9]

Broman and F

E.I. Broman and F. Camia. Universal behavior of connectivity properties in fractal percolation models. Electron. J. Probab., 15:1394–1414, 2010

2010
[10]

Brydges, J

D.C. Brydges, J. Fr¨ ohlich, A.D. Sokal, The random-walk representation of classical spin systems and correlation inequalities. II. The skeleton inequalities, Comm. Math. Phys. 91: 117–139 1983

1983
[11]

Brydges, J

D.C. Brydges, J. Fr¨ ohlich, T. Spencer, The random walk representation of classical spin systems and correlation inequalities, Comm. Math. Phys. , 83: 123–150, 1982

1982
[12]

van de Brug, F

T. van de Brug, F. Camia, and M. Lis. Spin systems from loop soups. Electron. J. Probab., 23:17 pp., 2018. – 63 –

2018
[13]

F. Camia. Scaling Limits, Brownian Loops and Conformal Fields. In Advances in Disordered Systems, Random Processes and Some Applications : 205–269. Cambridge Univ. Press, Cambridge, 2017

2017
[14]

Camia, V.F

F. Camia, V.F. Fiot, A. Gandolﬁ, and M. Kleban. Exact Correlation Functions in the Brownian Loop Soup. Preprint arXiv:1912.00973, 2019

1912 arXiv
[16]

Camia, C

F. Camia, C. Garban, and C.M. Newman. Planar Ising magnetization ﬁeld I. Uniqueness of the critical scaling limit. Ann. Probab. 43, 528–571, 2015

2015
[17]

Camia, C

F. Camia, C. Garban, and C.M. Newman. Planar Ising magnetization ﬁeld II. Properties of the critical and near-critical scaling limits. Ann. Inst. H. Poincar´ e Probab. Statist. 52, 146–161, 2016

2016
[18]

Carpentier and P

D. Carpentier and P. Le Doussal Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh-Gordon models. Phys.Rev. E 63:026110, 2001

2001
[19]

David, A

F. David, A. Kupiainen, R. Rhodes, and V. Vargas Liouville Quantum Gravity on the Riemann sphere. C ommun. Math. Phys. 342 (3): 869–907, 2016

2016
[20]

Di Francesco, P

P. Di Francesco, P. Mathieu, and D. S´ en´ echal.Conformal ﬁeld theory . Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997

1997
[21]

Dynkin, Markov processes as a tool in ﬁeld theory, J

E.B. Dynkin, Markov processes as a tool in ﬁeld theory, J. Funct. Anal. 50: 167–187, 1983

1983
[22]

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

1984
[23]

Duplantier and S

B. Duplantier and S. Sheﬃeld. Liouville quantum gravity and KPZ. Inventiones Mathematicae, 185(2):333–393, 2011

2011
[24]

Fern´ andez, J

R. Fern´ andez, J. Fr¨ ohlich, A.D. Sokal,Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory , Springer-Verlag 1992

1992
[26]

Gamsa and J

A. Gamsa and J. Cardy. Correlation functions of twist operators applied to single self-avoiding loops. J. Phys. A , (39):12983, 2006

2006
[27]

D. Grieser. Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Communications in Partial Diﬀerential Equations , 27(7-8):1283–1299, 2002

2002
[28]

Y. Han, Y. Wang, and M. Zinsmeister. On The Brownian Loop Measure. J. Stat. Phys. (175):987–1005, 2019. – 64 –

2019
[29]

M. Henkel. Conformal invariance and critical phenomena . Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999

1999
[30]

Junnila, E

J. Junnila, E. Saksman, and C. Webb. Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model. Preprint arXiv:1806.02118, 2018

2018 arXiv
[31]

J.P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Qu´ ebec, 9(2):105–150, 1985

1985
[32]

Kupiainen, R

A. Kupiainen, R. Rhodes, and V. Vargas Integrability of Liouville theory: proof of the DOZZ Formula. Preprint arXiv:1707.08785. 2017

2017 arXiv
[33]

Lacoin, R

H. Lacoin, R. Rhodes, and V. Vargas. Complex Gaussian multiplicative chaos. Comm. Math. Phys., 337(2):569–632, 2015

2015
[34]

Lambert, D

G. Lambert, D. Ostrovsky, and N. Simm. Subcritical multiplicative chaos for regularized counting statistics from random matrix theory. Comm. Math. Phys. , 360(1):1–54, 2018

2018
[35]

G. Last. Stochastic analysis for Poisson processes. In Stochastic Analysis for Poisson Point Processes, volume 7 of Bocconi & Springer Ser. : 1–36. Springer International Publishing, 2016

2016
[36]

Last and M

G. Last and M. Penrose. Lectures on the Poisson process, volume 7 of Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge, 2018

2018
[37]

Last and M

G. Last and M. Penrose. Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Related Fields, 150(3-4):663–690, 2011

2011
[38]

G.F. Lawler. Conformally invariant processes in the plane , volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005

2005
[39]

Lawler and W

G.F. Lawler and W. Werner. The Brownian loop soup. Probab. Theory and Related Fields, 128(4):565–588, 2004

2004
[40]

Le Jan, Markov loops and renormalization, Ann

Y. Le Jan, Markov loops and renormalization, Ann. Probab. 38: 1280–1319, 2010

2010
[41]

Y. Le Jan. Markov paths, loops and ﬁelds , in Lecture Notes in Mathematics, volume 2026, Ecole d’Et´ e de Probabilit´ e de St. Flour, Springer, Berlin, 2012

2026
[42]

Y. Le Jan. Brownian winding ﬁelds. arXiv preprint arXiv:1811.02737 , 2018

2018 arXiv
[43]

Mandelbrot

B. Mandelbrot. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier Journal of Fluid Mechanics 62.2:331–358, 1974

1974
[44]

Miller and S

J. Miller and S. Sheﬃeld, Imaginary geometry I: interacting SLEs. Probab. Theory and Related Fields 164. no. 3-4, 553–705, 2016

2016
[45]

Nacu and W

S. Nacu and W. Werner. Random soups, carpets and fractal dimensions. Journal of the London Mathematical Society 83, Issue 3: 789–809, 2011

2011
[46]

Nikula, E

M. Nikula, E. Saksman, and C. Webb. Multiplicative chaos and the characteristic polynomial of the cue: the L1-phase. Preprint arXiv:1806.01831, 2018. – 65 –

2018 arXiv
[47]

Nourdin and G

I. Nourdin and G. Peccati. Normal approximations with Malliavin calculus: From Stein’s method to universality. , volume 192 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2012

2012
[48]

Peccati and M.S

G. Peccati and M.S. Taqqu. Wiener chaos: moments, cumulants and diagrams: A survey with computer implementation, (supplementary material available online). , Volume 1 of Bocconi & Springer Series . Springer, Milan; Bocconi University Press, Milan, 2011

2011
[49]

Polyakov

A.M. Polyakov. Conformal symmetry of critical ﬂuctuations. JETP Lett., 12:381–383, 1970

1970
[50]

Pommerenke

Ch. Pommerenke. Boundary behaviour of conformal maps , Volume 299 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1992

1992
[51]

Rhodes and V

R. Rhodes and V. Vargas. Gaussian multiplicative chaos and Liouville quantum gravity. In Stochastic processes and random matrices, pages 548–577. Oxford Univ. Press, Oxford, 2017

2017
[52]

Robert and V

R. Robert and V. Vargas. Gaussian multiplicative chaos revisited. Ann. Probab., 38(2):605–631, 2010

2010
[53]

Saksman and C

E. Saksman and C. Webb. Multiplicative chaos measures for a random model of the Riemann zeta function. arXiv preprint arXiv:1604.08378 , 2016

2016 arXiv
[54]

A. Shamov. On Gaussian multiplicative chaos. J. Funct. Anal., 270(9):3224–3261, 2016

2016
[55]

Sheﬃeld, Exploration trees and conformal loop ensembles, Duke Math

S. Sheﬃeld, Exploration trees and conformal loop ensembles, Duke Math. J. 147: 79–129, 2009

2009
[56]

Sheﬃeld Conformal weldings of random surfaces: SLE and the quantum gravity zipper

S. Sheﬃeld Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann.Probab. 44(5), 3474–3545: 2016

2016
[57]

Sheﬃeld and W

S. Sheﬃeld and W. Werner, Conformal Loop Ensembles: the Markovian characterization and the loop-soup construction, Ann. Math. 176: 1827-1917, 2012

1917
[58]

D.W. Stroock. Homogeneous chaos revisited. In S´ eminaire de Probabilit´ es, XXI, volume 1247 of Lecture Notes in Math. , pages 1–7. Springer, Berlin, 1987

1987
[59]

Surgailis

D. Surgailis. Zones of attraction of self-similar multiple integrals. Lithuanian Math. Journal, 22(3):327–340, 1982

1982
[60]

Enrico Fermi,

K. Symanzik. Euclidean quantum ﬁeld theory. In Local quantum theory, Proceedings of the International School of Physics “Enrico Fermi,” course 45 (R. Jost editor), pp. 152–223, Academic Press, New York, 1969

1969
[61]

Sznitman

A.S. Sznitman. Topics in Occupation Times and Gaussian Free Field , Z¨ urich Lectures in Advanced Mathematics, European Mathematical Society Publishing House, Z¨ urich, 2012

2012
[62]

C. Webb. The characteristic polynomial of a random unitary matrix and Gaussian – 66 – multiplicative chaos—theL2-phase. Electron. J. Probab., 20:no. 104, 21, 2015

2015
[63]

W. Werner. SLEs as boundaries of clusters of Brownian loops, C. R. Acad. Sci.–Ser. I–Math. 337: 481–486, 2003

2003
[64]

W. Werner. Some recent aspects of random conformally invariant systems, in Les Houches Scool Proceedings: Session LXXXII, Mathematical Statistical Physics (A. Bovier, F. Dunlop, A. van Enter, J. Dalibard editors), pp. 57–98, Elsevier, 2006

2006
[65]

W. Werner. The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc., 21(1):137–169, 2008

2008
[66]

H. Weyl. ¨Uber die asymptotische Verteilung der Eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 1911:110–117, 1911. – 67 –

1911

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [15]

Camia, A

F. Camia, A. Gandolﬁ, and M. Kleban. Conformal correlation functions in the Brownian loop soup. Nuclear Physics B , 902:483–507, 2016

work page 2016

[2] [25]

Freivogel and M

B. Freivogel and M. Kleban. A conformal ﬁeld theory for eternal inﬂation ? J. High Energy Phys., (12):019, 31, 2009

work page 2009

[3] [1]

Aizenman

M. Aizenman. Geometric Analysis of φ4 Fields and Ising Models. Parts I and II, Comm. Math. Phys. 86: 1–48, 1982

work page 1982

[4] [2]

Astala, P

K. Astala, P. Jones, A. Kupiainen, and E. Saksman. Random conformal weldings. Acta Math. 207 , no. 2: 203–254, 2011

work page 2011

[5] [3]

Belavin, A.M

A.A. Belavin, A.M. Polyakov, and A.B. Zamolodchikov. Inﬁnite conformal symmetry in two-dimensional quantum ﬁeld theory. Nuclear Phys. B , 241(2):333–380, 1984

work page 1984

[6] [4]

Belavin, A.M

A.A. Belavin, A.M. Polyakov, and A.B. Zamolodchikov. Inﬁnite conformal symmetry of critical ﬂuctuations in two dimensions. J. Statist. Phys. , 34(5-6):763–774, 1984

work page 1984

[7] [5]

Berestycki

N. Berestycki. An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab., 22:Paper No. 27, 12, 2017

work page 2017

[8] [6]

Berestycki, C

N. Berestycki, C. Webb, and M. D. Wong. Random Hermitian matrices and Gaussian multiplicative chaos. Probab. Theory Related Fields, 172(1-2):103–189, 2018

work page 2018