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L2 limits of spherical Gaussian multiplicative chaos test functionals

Statement

Assume the covariance and smoothing conventions of the spherical Gaussian free field and its normalized chaos regularization defined earlier on this page. For $0 < γ < \sqrt{2}$ and every real continuous $f$ on $S$ , the scalar random variables

$M_{γ, ε} (f) : = \int_{S} f (z) \exp ⁣ (γ X_{ε} (z) - \frac{γ^{2}}{2} E [X_{ε} (z)^{2}]) d A_{0} (z)$

converge in $L^{2} (Ω, P)$ as $ε ↓ 0$ . Their limit is unchanged when one smoothing family is replaced by another. For real continuous $f, h$ , the limiting functionals satisfy

$E [M_{γ} (f)] = \int_{S} f d A_{0}, E [M_{γ} (f) M_{γ} (h)] = \int_{S} \int_{S} f (z) h (w) e^{γ^{2} K (z, w)} d A_{0} (z) d A_{0} (w) .$

The statement concerns scalar test-function limits; it does not assert a Radon-measure-valued version for the full range $0 < γ < 2$ .

Statement references

Gaussian free field on the Riemann sphere in the Liouville convention

Spherical Gaussian multiplicative chaos interaction data

Facts & Assumptions

Given: $X$ is a centered Gaussian random distribution on the compact sphere, $K$ has the logarithmic singularity specified by the GFF definition, and $M_{γ, ε}$ is its normalized regularization.

[F1]

Gaussian free field on the Riemann sphere in the Liouville convention: the covariance is $K (z, w) = \log (1 / ∣ z - w ∣) + \log ∣ z ∣_{+} + \log ∣ w ∣_{+}$ , paired on tests against the smooth round area measure $d A_{r d}$ ; the GMC integration measure remains $d A_{0}$ ; every pair of admitted smoothings has mixed covariance converging off the diagonal and bounded above by $C + \log^{+} (1 / (d_{r d} (z, w) + ε + δ))$ uniformly.

[F2]

Spherical Gaussian multiplicative chaos interaction data: $M_{γ, ε} (d z) = e^{γ X_{ε} (z) - γ^{2} E [X_{ε} (z)^{2}] / 2} d A_{0} (z)$ .

[A1]

$0 < γ < \sqrt{2}$ , so $γ^{2} < 2$ .

Proof

1.1F1A1algebra

The covariance bound in [F1] implies $e^{γ^{2} K_{ε, δ} (z, w)} \leq C_{1} (d_{r d} (z, w)^{- γ^{2}} + 1)$ for a constant independent of the two smoothing parameters. The background density $d A_{0}$ is continuous and bounded in every compact chart, so near the diagonal its integral is bounded by a constant times $\int_{0}^{1} r^{1 - γ^{2}} d r = (2 - γ^{2})^{- 1} < \infty$ by [A1]; away from the diagonal it is bounded.

2.1F1F2step 1.1algebra

If $U, V$ are jointly centered Gaussian, their moment-generating function gives $E [e^{γ U - γ^{2} E [U^{2}] / 2} e^{γ V - γ^{2} E [V^{2}] / 2}] = e^{γ^{2} E [U V]}$ . Fubini applied to the absolutely integrable kernel bound in step 1.1 therefore gives $E [M_{γ, ε} (f) M_{γ, δ} (h)] = \int_{S} \int_{S} f (z) h (w) e^{γ^{2} K_{ε, δ} (z, w)} d A_{0} (z) d A_{0} (w)$ .

3.1F1step 1.1step 2.1algebra

The off-diagonal covariance convergence in [F1], the integrable bound in step 1.1, and dominated convergence show that the moment in step 2.1 tends to $\int_{S} \int_{S} f (z) h (w) e^{γ^{2} K (z, w)} d A_{0} (z) d A_{0} (w)$ as $ε, δ ↓ 0$ . Taking $h = 1$ in the same Gaussian exponential identity gives $E [M_{γ, ε} (f)] = \int_{S} f d A_{0}$ for every $ε$ .

4.1step 2.1step 3.1algebra

Expand $E [∣ M_{γ, ε} (f) - M_{γ, δ} (f) ∣^{2}]$ into two self terms and two mixed terms. By step 3.1 each term tends to $I_{f}$ with its indicated sign, so the limit is $I_{f} + I_{f} - I_{f} - I_{f} = 0$ . Thus the regularized scalar functionals are Cauchy in the complete space $L^{2} (Ω, P)$ and have an $L^{2}$ limit.

5.1F1step 3.1step 4.1∎

For two smoothing families, the same uniform covariance bound and off-diagonal limit in [F1] give the same mixed second-moment limit in step 3.1. Repeating the four-term difference calculation of step 4.1 shows that their scalar limits agree. The limits have the first and second moments asserted in the Statement by step 3.1.

Remarks

The proof stops at scalar $L^{2}$ limits. Turning the simultaneous family of limits into a finite Radon random measure, and extending existence and mollifier-independence to $\sqrt{2} \leq γ < 2$ , require additional arguments. The cited lecture notes prove the L2 case and refer elsewhere for the remaining subcritical interval; they are not treated as supplying those local arguments.

Depends on

Dependency tree · 2 levels

Sources

Rémi Rhodes and Vincent Vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, arXiv:1602.07323v1