Critical Non-Gaussian Multiplicative Chaos

• Critical non-Gaussian multiplicative chaos is defined as the scaling limit at the critical parameter (γ = 2) from non-Gaussian inputs such as random zeta and Brownian local time models.
• Key constructions use a Gaussian approximation in the random zeta model and continuity plus stochastic-calculus methods in the Brownian setting to achieve convergence via critical renormalizations.
• The critical regime requires adjusted normalization methods, yielding universal multifractal random measures with distinct phase transitions and conformal covariance properties.

to=arxiv_search.^^^^^^^^84query84^^^^^^^^ 菲律宾申博json {"^^^^^^^^84query84 non-Gaussian multiplicative chaos\"^^^^^^^^84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^\" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" to=arxiv_search.^^^^^^^^84query84^^^^^^^^ code d天天 [{"^^^^^^^^84arxiv_id84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84query84^^^^^^^^" OR ti:\84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84relevance84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:4(Jego, 2020)84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84version84^^^^^^^^"] OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84sort_by84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84arxiv_id84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84version84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^ Hu and Shi in ^^^^84 OR ti:\84query84 OR ti:\84query84^^^^."^^^^^^^^84sort_by84query84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" uniqueness of the Gaussian multiplicative chaos revisited"^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" Junnila"^^^^^^^^84sort_by84 OR ti:4^^^^^^^^" OR ti:4(Jego, 2020)84^^^^^^^^"] OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84query84all:\84sort_by84^^^^-^^^^84query84relevance84^^^^- OR ti:\84^^^^^^^^" multiplicative chaos measures are random measures obtained by exponentiating approximation sequences of Gaussian fields whose covariance functions have the same logarithmic singularity structure. A uniqueness result by Shamov states that the law of the random measure depends only on the limiting covariance and not on the particular approximation sequence^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ provided some mild assumptions hold. We prove a more general uniqueness result^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ improve and simplify some of the proofs^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and introduce a related convergence theorem which is particularly useful in applications to non-Gaussian multiplicative chaos. As a by-product we give a short proof of a result by Berestycki."^^^^^^^^84sort_by84query84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" deviations of the derivative martingale in the branching random walk"^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" Z. Jibraeil"^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^"Yujin H. Kim"^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^"Pierre Mallein"]^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84query84 OR ti:\84query84^^^^-^^^^84query84relevance84^^^^- OR ti:\84^^^^^^^^" study the upper and lower tails of the derivative martingale in the branching random walk. Under minimal integrability assumptions^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ we prove that the tail distribution of the derivative martingale is regularly varying with index -^^^^84all:\84 OR ti:\84^^^^^^^^ identifying explicitly the leading constant. The proof is based on a new decomposition of the derivative martingale using stopping lines."}] to=arxiv_search.^^^^^^^^84query84^^^^^^^^ 天天爱彩票网站json {"^^^^^^^^84query84 multiplicative chaos\" AND au^^^^^^^^84sort_by84all:\84 OR ^^^^^^^^84all:\84 Rudd\""^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" to=arxiv_search.^^^^^^^^84query84^^^^^^^^ code 微信天天中彩票 [{"^^^^^^^^84arxiv_id84 OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" measures^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ introduced in [Jeg^^^^84 OR ti:\84query84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ AHS^^^^84 OR ti:\84query84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ BBK^^^^84version84max_results84^^^^] OR ti:\84^^^^^^^^ are random Borel measures that can be formally defined by exponentiating PRESERVED_PLACEHOLDER_^^^^84max_results84^^^^ times the square root of the local times of planar Brownian motion. So far^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ only the subcritical measures where the parameter PRESERVED_PLACEHOLDER_^^^^84sort_by84^^^^ is less than ^^^^84 OR ti:\84^^^^ were studied. This article considers the critical case where PRESERVED_PLACEHOLDER_^^^^84relevance84 OR ti:\84^^^^^^^^ using three different approximation procedures which all lead to the same universal measure. On the one hand^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ we exponentiate the square root of the local times of small circles and show convergence in the Seneta--Heyde normalisation as well as in the derivative martingale normalisation. On the other hand^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ but there are essential differences^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the main one being the lack of Gaussianity which prevents the use of Kahane's inequality and hence a priori controls. Instead^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute."^^^^^^^^84sort_by84query84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84arxiv_id84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84version84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" OR ti:\84^^^^^^^^" this article^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the author constructs a family of random Borel measures associated to planar Brownian motion. These measures are built in a similar fashion to Gaussian multiplicative chaos^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ but^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ instead of a Gaussian process^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ one uses the local times of the Brownian circles. One obtains one universal measure for each parameter PRESERVED_PLACEHOLDER_^^^^84arxiv_id84^^^^ by considering several approximation processes. This construction extends previous works of Bass^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ Burdzy and Khoshnevisan in ^^^^84all:\84version84version84max_results84 OR ti:\84^^^^^^^^ and A\\"idekon^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ Hu and Shi in ^^^^84 OR ti:\84query84 OR ti:\84query84^^^^."}] Critical non-Gaussian multiplicative chaos refers^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ in the present arXiv literature^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ to multiplicative-chaos limits obtained at the critical parameter from non-Gaussian inputs. Two explicit constructions organize the subject. In the random zeta model of Saksman and Webb^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ a stochastic approximation of powers of the Riemann zeta function converges to a non-Gaussian multiplicative chaos measure^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ with results covering both the subcritical and critical regimes ^^^^^^^^84 OR ti:\84 OR ti:\84^^^^^^^^. In ^^^^^^^^84max_results84all:\84 OR ti:\84^^^^^^^^ ^^^^^^^^84all:\84max_results84^^^^^^^^ studies the case PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84^^^^ for measures built from the square root of Brownian circle local times^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ proving that three different approximation procedures lead to the same universal critical measure; this is described as the first example of a non-Gaussian critical multiplicative chaos ^^^^^^^^84(Jego, 2020)84

^^^^84all:\84^^^^. Criticality and the multiplicative-chaos framework

In both principal examples^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the construction begins with a family of positive random measures indexed by a parameter PRESERVED_PLACEHOLDER_^^^^84version84 OR ti:\84^^^^^^^^ often interpreted as a temperature parameter. The critical value is PRESERVED_PLACEHOLDER_^^^^84all:\84query84^^^^.

For the random zeta model^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the approximating measures are

PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84^^^^

and the critical parameter is explicitly PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84^^^^. For every PRESERVED_PLACEHOLDER_^^^^84all:\84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ PRESERVED_PLACEHOLDER_^^^^84all:\84max_results84^^^^ converges almost surely^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ with respect to the weak topology on Radon measures^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ to a nontrivial limiting random measure PRESERVED_PLACEHOLDER_^^^^84all:\84sort_by84^^^^. At PRESERVED_PLACEHOLDER_^^^^84all:\84relevance84 OR ti:\84^^^^^^^^ one must pass to a critical normalization^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84all:\84arxiv_id84^^^^

which converges in distribution^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ again in the weak topology^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ to a nontrivial random measure PRESERVED_PLACEHOLDER_^^^^84all:\84(Jego, 2020)84^^^^ ^^^^^^^^84 OR ti:\84 OR ti:\84^^^^^^^^.

For ^^^^^^^^84max_results84all:\84 OR ti:\84^^^^^^^^ one starts from the subcritical regularization

PRESERVED_PLACEHOLDER_^^^^84all:\84version84^^^^

where PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84query84^^^^ is the PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84all:\84^^^^-circle local time of planar Brownian motion. For PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84 OR ti:\84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^ converges in probability to a nontrivial random Borel measure PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84max_results84^^^^. At PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84sort_by84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the naively normalized PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84relevance84^^^^ vanishes^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ so one introduces critical renormalizations instead. Jego proves that both Seneta–Heyde and derivative-martingale normalizations converge in probability to the same nondegenerate random measure PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84arxiv_id84^^^^ ^^^^^^^^84(Jego, 2020)84

These two settings already show the basic meaning of criticality in the non-Gaussian case^^^^^^^^84sort_by84all:\84^^^^^^^^ the direct exponential normalization becomes degenerate at PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84(Jego, 2020)84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ but an adjusted critical renormalization yields a nontrivial limit.

^^^^84 OR ti:\84^^^^. The random zeta model and its critical chaos

The zeta-model construction uses the primes PRESERVED_PLACEHOLDER_^^^^84 OR ti:\84version84^^^^ and i.i.d. random phases PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84query84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ uniform on PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84all:\84^^^^. The field on PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84 OR ti:\84^^^^ is

PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^

The associated measure PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84max_results84^^^^ is positive on PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84sort_by84^^^^ and has total mass PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84relevance84^^^^ in expectation ^^^^^^^^84 OR ti:\84 OR ti:\84^^^^^^^^.

A central structural fact is the decomposition

PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84arxiv_id84^^^^

where PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84(Jego, 2020)84^^^^ is a log-correlated Gaussian field and PRESERVED_PLACEHOLDER_^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84version84^^^^ converges almost surely in PRESERVED_PLACEHOLDER_^^^^84max_results84query84^^^^ to a continuous limit PRESERVED_PLACEHOLDER_^^^^84max_results84all:\84^^^^. Concretely^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84max_results84 OR ti:\84^^^^

with PRESERVED_PLACEHOLDER_^^^^84max_results84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^ i.i.d. PRESERVED_PLACEHOLDER_^^^^84max_results84max_results84 OR ti:\84^^^^^^^^ while the error satisfies uniform exponential integrability^^^^^^^^84sort_by84all:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84max_results84sort_by84^^^^

The convergence theory splits into the usual subcritical^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ critical^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and supercritical regimes. For every PRESERVED_PLACEHOLDER_^^^^84max_results84relevance84 OR ti:\84^^^^^^^^ the measures PRESERVED_PLACEHOLDER_^^^^84max_results84arxiv_id84^^^^ converge almost surely to a nontrivial PRESERVED_PLACEHOLDER_^^^^84max_results84(Jego, 2020)84 OR ti:\84^^^^^^^^ and for each PRESERVED_PLACEHOLDER_^^^^84max_results84version84^^^^ one has

PRESERVED_PLACEHOLDER_^^^^84sort_by84query84^^^^

At criticality^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the normalized measures PRESERVED_PLACEHOLDER_^^^^84sort_by84all:\84^^^^ converge in distribution to a nontrivial limit PRESERVED_PLACEHOLDER_^^^^84sort_by84 OR ti:\84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and for every PRESERVED_PLACEHOLDER_^^^^84sort_by84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84sort_by84max_results84^^^^

For PRESERVED_PLACEHOLDER_^^^^84sort_by84sort_by84 OR ti:\84^^^^^^^^ the limit is PRESERVED_PLACEHOLDER_^^^^84sort_by84relevance84^^^^.

The limiting subcritical measure is also a nontrivial multifractal random measure. If PRESERVED_PLACEHOLDER_^^^^84sort_by84arxiv_id84^^^^ denotes the interval of length PRESERVED_PLACEHOLDER_^^^^84sort_by84(Jego, 2020)84^^^^ around PRESERVED_PLACEHOLDER_^^^^84sort_by84version84 OR ti:\84^^^^^^^^ then for PRESERVED_PLACEHOLDER_^^^^84relevance84query84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84relevance84all:\84^^^^

Accordingly^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the scaling spectrum is

PRESERVED_PLACEHOLDER_^^^^84relevance84 OR ti:\84^^^^

The paper also records tail asymptotics inherited from standard Gaussian-chaos results^^^^^^^^84sort_by84all:\84^^^^^^^^ for PRESERVED_PLACEHOLDER_^^^^84relevance84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^ and large PRESERVED_PLACEHOLDER_^^^^84relevance84max_results84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84relevance84sort_by84^^^^

^^^^84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^. ^^^^^^^^84all:\84 OR ti:\84^^^^^^^^

The Brownian construction is formulated in a bounded simply-connected domain PRESERVED_PLACEHOLDER_^^^^84relevance84relevance84 OR ti:\84^^^^^^^^ with planar Brownian motion PRESERVED_PLACEHOLDER_^^^^84relevance84arxiv_id84^^^^ started at PRESERVED_PLACEHOLDER_^^^^84relevance84(Jego, 2020)84^^^^ and stopped at

PRESERVED_PLACEHOLDER_^^^^84relevance84version84^^^^

For PRESERVED_PLACEHOLDER_^^^^84arxiv_id84query84^^^^ and small PRESERVED_PLACEHOLDER_^^^^84arxiv_id84all:\84 OR ti:\84^^^^^^^^ the PRESERVED_PLACEHOLDER_^^^^84arxiv_id84 OR ti:\84^^^^-circle local time is

PRESERVED_PLACEHOLDER_^^^^84arxiv_id84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^

and this exists jointly in PRESERVED_PLACEHOLDER_^^^^84arxiv_id84max_results84^^^^ ^^^^^^^^84(Jego, 2020)84

The subcritical ^^^^^^^^84max_results84all:\84^^^^^^^^ is

PRESERVED_PLACEHOLDER_^^^^84arxiv_id84sort_by84^^^^

with convergence in probability

PRESERVED_PLACEHOLDER_^^^^84arxiv_id84relevance84^^^^

The critical regime PRESERVED_PLACEHOLDER_^^^^84arxiv_id84arxiv_id84^^^^ requires renormalization. Jego proves convergence for two distinct critical schemes^^^^^^^^84sort_by84all:\84^^^^^^^^

• Seneta–Heyde normalization^^^^^^^^84sort_by84all:\84

PRESERVED_PLACEHOLDER_^^^^84arxiv_id84(Jego, 2020)84^^^^

• Derivative-martingale normalization^^^^^^^^84sort_by84all:\84

PRESERVED_PLACEHOLDER_^^^^84arxiv_id84version84^^^^

Both converge in probability^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ for the weak topology^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ to the same random measure PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84query84^^^^. Moreover^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84all:\84^^^^

and PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84 OR ti:\84^^^^ has no atoms. The paper also proves a subcritical-to-critical limit^^^^^^^^84sort_by84all:\84^^^^^^^^ one may choose a modification of PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^ such that for each fixed continuous PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84max_results84 OR ti:\84^^^^^^^^ the map PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84sort_by84^^^^ is lower-semicontinuous on PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84relevance84 OR ti:\84^^^^^^^^ and for Borel PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84arxiv_id84 OR ti:\84^^^^^^^^

PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84(Jego, 2020)84^^^^

A further structural property is conformal covariance. If PRESERVED_PLACEHOLDER_^^^^84(Jego, 2020)84version84^^^^ is conformal^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ then the corresponding critical measures satisfy in law

PRESERVED_PLACEHOLDER_^^^^84version84query84^^^^

This places the critical Brownian measure close to the conformally covariant objects familiar in planar probabilistic field theory^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ while remaining genuinely non-Gaussian ^^^^^^^^84(Jego, 2020)84

^^^^84max_results84^^^^. Normalizations^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ universality^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and uniqueness

A defining feature of critical multiplicative chaos is that multiple renormalization procedures can converge to the same limiting object. In the Brownian setting^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ this is a theorem^^^^^^^^84sort_by84all:\84^^^^^^^^ Seneta–Heyde normalization^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ derivative-martingale normalization^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and the subcritical limit all yield the same law for the critical measure PRESERVED_PLACEHOLDER_^^^^84version84all:\84^^^^ ^^^^^^^^84(Jego, 2020)84 The resulting universality statement is not merely qualitative; it is encoded in the convergence

PRESERVED_PLACEHOLDER_^^^^84version84 OR ti:\84^^^^

which identifies the critical measure as the endpoint of the subcritical family.

In the random zeta model^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the comparison principle is formulated differently. One introduces the Gaussian multiplicative-chaos approximants

PRESERVED_PLACEHOLDER_^^^^84version84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^

where PRESERVED_PLACEHOLDER_^^^^84version84max_results84^^^^ is the Gaussian part of PRESERVED_PLACEHOLDER_^^^^84version84sort_by84^^^^. By Kahane’s theory^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ for PRESERVED_PLACEHOLDER_^^^^84version84relevance84 OR ti:\84^^^^^^^^ PRESERVED_PLACEHOLDER_^^^^84version84arxiv_id84^^^^ almost surely^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ whereas for PRESERVED_PLACEHOLDER_^^^^84version84(Jego, 2020)84^^^^ the Gaussian chaos either degenerates or requires critical normalization. The proof strategy then uses the uniform control of PRESERVED_PLACEHOLDER_^^^^84version84version84^^^^ to show that PRESERVED_PLACEHOLDER_^^^^84all:\84query84query84^^^^ and its critical analogue differ from PRESERVED_PLACEHOLDER_^^^^84all:\84query84all:\84^^^^ only by multiplicative factors PRESERVED_PLACEHOLDER_^^^^84all:\84query84 OR ti:\84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ which gives tightness in the subcritical case and convergence in the critical case ^^^^^^^^84 OR ti:\84 OR ti:\84^^^^^^^^.

For the critical zeta model^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the limit is identified by a Gaussian-comparison uniqueness theorem. The paper explicitly invokes a theorem from Junnila and Saksman’s “The uniqueness of the Gaussian multiplicative chaos revisited” to identify the critical limit with the critical Gaussian chaos up to an independent random multiplier (&&&^^^^84sort_by84^^^^&&&). This does not make the zeta-model chaos Gaussian; rather^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ it shows that a non-Gaussian construction can be rigid enough to be pinned down through comparison with Gaussian multiplicative chaos.

^^^^84sort_by84^^^^. Analytic mechanisms beyond Gaussianity

The two main examples illustrate two different ways of handling the loss of Gaussian structure.

In the random zeta model^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the decisive ingredient is a “good” Gaussian approximation. The proof proceeds by decomposing PRESERVED_PLACEHOLDER_^^^^84all:\84query84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ establishing almost sure convergence of PRESERVED_PLACEHOLDER_^^^^84all:\84query84max_results84^^^^ in PRESERVED_PLACEHOLDER_^^^^84all:\84query84sort_by84 OR ti:\84^^^^^^^^ and proving uniform exponential-tail bounds for PRESERVED_PLACEHOLDER_^^^^84all:\84query84relevance84^^^^. A refined block-coupling proposition underlies this step^^^^^^^^84sort_by84all:\84^^^^^^^^ if PRESERVED_PLACEHOLDER_^^^^84all:\84query84arxiv_id84^^^^ are independent mean-zero PRESERVED_PLACEHOLDER_^^^^84all:\84query84(Jego, 2020)84^^^^-valued random vectors with PRESERVED_PLACEHOLDER_^^^^84all:\84query84version84 OR ti:\84^^^^^^^^ PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84query84 OR ti:\84^^^^^^^^ and PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84all:\84 OR ti:\84^^^^^^^^ then one can construct a Gaussian

PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84 OR ti:\84^^^^

such that the coupling error

PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^

satisfies

PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84max_results84^^^^

for some PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84sort_by84^^^^ ^^^^^^^^84 OR ti:\84 OR ti:\84^^^^^^^^. The method is therefore non-Gaussian at the level of the original field^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ but strongly Gaussian in its approximation theory.

In the Brownian setting^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the obstruction is more direct^^^^^^^^84sort_by84all:\84^^^^^^^^ PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84relevance84^^^^ is non-Gaussian^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ so Kahane’s inequalities are unavailable. Jego states this explicitly and replaces them with a continuity lemma that allows local times at nearby centers to be decoupled. The lemma controls conditional probabilities by

PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84arxiv_id84^^^^

with

PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84(Jego, 2020)84^^^^

This is combined with the Markovian representation of local times in terms of zero-dimensional Bessel processes through Ray–Knight^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ barrier estimates for Brownian motion and Bessel processes^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and second-moment arguments proving that truncated critical measures are uniformly PRESERVED_PLACEHOLDER_^^^^84all:\84all:\84version84^^^^-bounded and Cauchy in PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84query84^^^^ ^^^^^^^^84(Jego, 2020)84

A common misconception is that non-Gaussian multiplicative chaos must be treated without reference to Gaussian multiplicative chaos. The two papers show a more precise picture. In the zeta model^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ Gaussian comparison is central and technically effective; in the Brownian model^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ the main novelty is precisely that the Gaussian convexity toolkit fails and must be replaced by mixing^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ continuity^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and Bessel-process estimates.

^^^^84relevance84^^^^. Structural properties and mathematical significance

The zeta-model limit exhibits the standard multiplicative-chaos signatures of phase transition^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ moment thresholds^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and multifractality. The threshold PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84all:\84^^^^ governs moment finiteness and the multifractal spectrum

PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84 OR ti:\84^^^^

The limiting measure is therefore not merely nontrivial; it is a nontrivial multifractal random measure ^^^^^^^^84 OR ti:\84 OR ti:\84^^^^^^^^.

The Brownian critical measure is distinguished by a different set of structural properties^^^^^^^^84sort_by84all:\84^^^^^^^^ positivity^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ nonatomicity^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ infinite first moment of the total mass^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ and conformal covariance with exponent PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84 OR id:(Jego, 2020) OR id:(Saksman et al., 2016)84^^^^. The coexistence of PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84max_results84^^^^ almost surely and PRESERVED_PLACEHOLDER_^^^^84all:\84 OR ti:\84sort_by84^^^^ is one of the characteristic critical phenomena established in the paper ^^^^^^^^84(Jego, 2020)84

Taken together^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ these examples delimit the present mathematical meaning of critical non-Gaussian multiplicative chaos. One model arises from a stochastic approximation to powers of the Riemann zeta function and is controlled by a precise Gaussian approximation; the other arises from Brownian circle local times and requires a substitute for Kahane’s convexity based on continuity and stochastic-calculus methods. A plausible implication is that critical non-Gaussian chaos should be viewed less as a single construction than as a critical universality mechanism^^^^^^^^84sort_by84all:\84^^^^^^^^ different non-Gaussian inputs can lead^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ after model-specific renormalization^^^^^^^^84sort_by84 OR ti:\84^^^^^^^^ to nondegenerate critical random measures with the same kinds of phase transition and normalization phenomena familiar from critical Gaussian multiplicative chaos.