Local disk GMC and uniform anchored inverse moments

D1. Elementary covariance estimates and L² existence. For the fixed compact convolution regularizations and circle averages, the joint regularized covariance satisfies $K_{\epsilon ,\eta }(z,w)\le \log (1\mathrm{/}(\mathrm{\mid }z-w\mathrm{\mid }\vee \epsilon \vee \eta ))+C\le \log (1\mathrm{/}\mathrm{\mid }z-w\mathrm{\mid })+C$ and converges to $K(z,w)$ off the diagonal. To prove the bound, rescale by the larger smoothing radius when $\mathrm{\mid }z-w\mathrm{\mid }$ is at most a fixed multiple of it; the averaged logarithm on a bounded disk is uniformly bounded after subtracting the logarithm of that radius. At larger separation, the logarithm differs from $\log (1\mathrm{/}\mathrm{\mid }z-w\mathrm{\mid })$ by a bounded amount. The bounded covariance remainder only changes $C$, and the mean-value identity gives the same estimate for circle averages.

For every bounded compactly supported test $f$, the Gaussian exponential identity gives $\mathbb{E}[M_{\epsilon }(f)M_{\eta }(f)]=\int f(z)f(w)e^{{\gamma }^2{K}_{\epsilon ,\eta }(z,w)}dzdw$, with any fixed bounded positive background densities included in the integrand.

Since $\mathrm{\mid }z-w{\mathrm{\mid }}^{-{\gamma }^2}$ is integrable on a bounded planar product exactly when ${\gamma }^2<2$, dominated convergence makes the second moments in step 2.1 Cauchy in $L^2$. Positivity and simultaneous convergence on a countable dense family then construct a finite positive Radon measure by the positive-functional representation supplied through the circle-GMC item [F3]. Cross-regularization $L^2$ convergence proves independence within the stated smoothing class, without assuming an external GMC limit theorem. The mean measure is the background area density.

D2. Fractional local scaling. Use the Gaussian interpolation and comparison proved in part C1 of the directly declared circle-GMC supplier [F3]. On a disk $B_r$, put $z=z_0+ru$ and compare $X_{\epsilon }(z_0+ru)$, for $\epsilon \le r$, with $Z_r+X_{\epsilon \mathrm{/}r}^{\mathrm{\prime }}(u)$, where $Z_r$ is an independent centered Gaussian of variance $\log (1\mathrm{/}r)$ and $X^{\mathrm{\prime }}$ is an independent copy of a fixed local logarithmic field with bounded continuous covariance remainder.

Scaling the logarithmic singularity shows that the covariance entries in this comparison differ by at most a fixed constant on the unit disk; the bounded continuous remainders only change that constant. Add a common Gaussian of sufficient variance to obtain the entrywise covariance order required by the comparison in [F3]. For $0<q<1$, concave comparison, extraction of the common Gaussian, and Jensen's inequality for the rescaled unit-disk mass give $\mathbb{E}[M_{\epsilon }(B_r{)}^q]\le C{r}^{2q}\exp ({\gamma }^2(q^2-q)\log (1\mathrm{/}r)\mathrm{/}2)=C{r}^{{\xi }_2(q)}$.

The background density contributes only fixed positive multiplicative bounds. For $q=1$ this is the mean measure estimate, ${\xi }_2(1)=2$. Passage to the L² limit gives the same bound for $M(B_r)$.

D3. Explicit independent-scale comparison field. Let B be the Euclidean unit disk and define kappa(x)=area(B intersect (B+x))/pi.

It is nonnegative, vanishes for $\mathrm{\mid }x\mathrm{\mid }\ge 2$, is positive definite (it is the autocorrelation of $1_B\mathrm{/}\sqrt{\pi }$), satisfies $\kappa (0)=1$ and $\mathrm{\mid }1-\kappa (x)\mathrm{\mid }\le C\mathrm{\mid }x\mathrm{\mid }$. Introduce real Gaussian white noise W on R² times (0,1), with control measure du dt/t, and put $Y_{ϵ}(z)={\pi }^{-1\mathrm{/}2}{\int }_{ϵ}^1{\int }_{{\mathbb{R}}^2}{t}^{-1}{1}_{\mathrm{\mid }u-z\mathrm{\mid }<t}W(du,dt\mathrm{/}t)$.

For a finite collection of z this Gaussian integral is the L² limit of step functions; consistency defines the continuous regularized field, or its measurable modification obtained by the usual Gaussian moment continuity bound. Its covariance is $C_{ϵ}(z,w)={\int }_{ϵ}^1\kappa ((z-w)\mathrm{/}t)dt\mathrm{/}t$.

The properties of kappa imply uniformly on bounded disks $C_{ϵ}(z,w)=\log (1\mathrm{/}(\mathrm{\mid }z-w\mathrm{\mid }\vee ϵ))+O(1)$.

Let ${\mathcal{F}}_n=\sigma (W\text{ restricted to cone-noise scales }t\ge 2^{-n})$. Increments on disjoint $t$-intervals are independent, so the normalized masses $M_n=Z_{2^{-n}}(D)$ form a nonnegative martingale with respect to $({\mathcal{F}}_n)$. The planar $L^2$ estimate in part D1 bounds ${\sup }_n\mathbb{E}{M}_n^2$. For $m\ge n$, $\mathbb{E}\mathrm{\mid }{M}_m-M_n{\mathrm{\mid }}^2=\mathbb{E}{M}_m^2-\mathbb{E}{M}_n^2$, since $\mathbb{E}[M_m\mid {\mathcal{F}}_n]=M_n$. The bounded increasing second moments therefore make $(M_n)$ Cauchy in $L^2$, with a limit $Z(D)$ and $\mathbb{E}Z(D)=\mathrm{area}(D)$. For almost-sure convergence, fix $n<N$ and let $\tau$ be the first $k\in (n,N]$ with $\mathrm{\mid }{M}_k-M_n\mathrm{\mid }\ge \lambda$. On each event $\{\tau =k\}\in {\mathcal{F}}_k$, $\mathbb{E}[M_N-M_k\mid {\mathcal{F}}_k]=0$, so expansion of the square gives $\mathbb{E}[\mathrm{\mid }{M}_N-M_n{\mathrm{\mid }}^2{\mathbf{1}}_{\{\tau =k\}}]\ge \mathbb{E}[\mathrm{\mid }{M}_k-M_n{\mathrm{\mid }}^2{\mathbf{1}}_{\{\tau =k\}}]$. Summing yields ${\lambda }^2\mathbb{P}({\max }_{n<k\le N}\mathrm{\mid }{M}_k-M_n\mathrm{\mid }\ge \lambda )\le \mathbb{E}\mathrm{\mid }{M}_N-M_n{\mathrm{\mid }}^2$. Since the second moments converge, choose $n_j$ so the right side is at most $2^{-4j}$ for every $N\ge n_j$ and put $\lambda =2^{-j}$. Letting $N\to \mathrm{\infty }$ bounds the tail crossing probability by $2^{-2j}$; the union bound on these tails tends to zero, so $(M_n)$ is almost surely Cauchy.

For each cutoff $\epsilon$, let ${\mathcal{F}}_{\epsilon }$ be the sigma-algebra generated by the cone white noise at scales $t\ge \epsilon$. Independent increments below scale $\epsilon$ give $Z_{\epsilon }(D)=\mathbb{E}[Z(D)\mid {\mathcal{F}}_{\epsilon }]$, first for dyadic cutoffs and then for arbitrary $\epsilon$ by the same conditional-independence identity. The mixed covariance formula and integrable bound of part D1 show that all intermediate scales converge in $L^2$ to the same $Z(D)$. This conditional-expectation identity controls negative moments of the regularizations; no uniform negative-moment limit is assumed.

D4. Negative moments of the cone limit. Fix one disk D and two separated subdisks of scale r inside it, with r sufficiently small. On their union the actual cone covariances are bounded above by those of two independent rescaled copies of the cone field on D plus a single independent common Gaussian of sufficiently large finite variance A. Within each subdisk this follows from $C(z,w)=\log (1\mathrm{/}\mathrm{\mid }z-w\mathrm{\mid })+O(1)$ and scaling; cross-covariances are bounded because the two disks are separated. The common variance may include log(1/r), and increasing its fixed value covers all bounded remainders. Convex comparison for exp(−t x), followed by the L² limit, gives ${\varphi }_D(t)\le E[{\varphi }_D(tB{)}^2]$, ${\varphi }_D(t)=E\exp (-tZ(D))$, $B=r^2\exp (\gamma G-{\gamma }^{2}A\mathrm{/}2)$, G Gaussian of variance A.

Here the model mass of each rescaled disk is B times an independent copy of Z(D); all scale factors have been kept. This is an inequality comparing Gaussian laws, not independence of the original subdisk masses.

The identical Laplace-recursion argument of the circle supplier now applies: taking t to infinity gives $a\le a^2$ for $a=P(Z(D)=0)$; the positive mean excludes $a=1$, so $a=0$. Splitting on $B<t^{-ϵ}$ yields ${\varphi }_D(\exp s)\le {\varphi }_D(\exp ((1-ϵ)s){)}^2+C\exp (-c{s}^2)$.

Choosing $1<\delta <2$ with $2(1-ϵ{)}^{\delta }>1$ and inducting on geometric s-intervals proves ${\varphi }_D(t)\le C\exp (-b(\log t{)}^{\delta })$. The gamma-integral identity then gives $E[Z(D{)}^{-a}]<\mathrm{\infty }$ for every $a>0$.

Conditional Jensen and part D3 give at every cutoff $E[Z_{ϵ}(D{)}^{-a}]\le E[Z(D{)}^{-a}]$.

Thus the cone regularizations already have the required uniform inverse moments.

D5. Transfer to the actual GFF and its actual regularizations. On any fixed compact anchor disk, part D1 and the cone covariance in D3 give $\mathrm{Cov}(X_{\epsilon }(z),X_{\epsilon }(w))\le C_{\epsilon }(z,w)+A_0$ for one $A_0$ independent of $\epsilon$.

Adding an independent common Gaussian of variance $A_0$ to $Y_{\epsilon }$ makes its covariance dominate the actual regularized covariance entrywise. The convex comparison for $F(x)=(x+\delta {)}^{-a}$, followed by monotone convergence as $\delta \downarrow 0$, gives $\mathbb{E}[M_{\epsilon }(D{)}^{-a}]\le e^{{\gamma }^2{A}_{0}a(a+1)\mathrm{/}2}\mathbb{E}[Z_{\epsilon }(D{)}^{-a}]\le C_a$.

A bounded positive background density is handled by its lower bound on D. Passing along an almost-sure subsequence of the L² mass convergence and using Fatou proves the corresponding inverse moments for M(D). This proves all of the interface, including uniformity before taking the regularization limit.