Steep Points of Gaussian Free Fields in Any Dimension

Abstract

This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets to a more general class of exceptional sets of a Gaussian free field (GFF). We adopt a circle or sphere averaging regularization to study a log-correlated or polynomial-correlated GFF in any dimension and introduce the notion of “f-steep points” of the GFF for a certain test function f. Roughly speaking, the f-steep points of the GFF are locations where, when weighted by the function f, the “rate of change” of the regularized field becomes unusually large. Different choices of f lead to the study of different exceptional behaviors of the GFF. We determine the Hausdorff dimension of the set consisting of f-steep points, from which not only can we recover the existing results on thick point sets for both log-correlated and polynomial-correlated GFFs, but we also obtain new results for exceptional sets that, to our best knowledge, have not been previously studied. Our method is inspired by the one used to study the thick point sets of the classical 2D log-correlated GFF.

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Notes

  1. 1.

    Similarly, the GFF is said to be polynomial-correlated when the corresponding Green function has a polynomial singularity. The rigorous definitions of these notions are given in Sect. 2.1.

  2. 2.

    The GFF associated with \(\Delta \) is referred to as a massless GFF, and the one associated with \(I-\Delta \) is a massive GFF. In \({\mathbb {R}}^{\nu }\) with \(\nu \ge 3\), we consider the massive GFF because the Green function of \(I-\Delta \) on \({\mathbb {R}}^{\nu }\) is technically more convenient to manipulate. See Remark 2.1 below.

  3. 3.

    In physics literature, the term “GFF” only refers to the case when \(p=1\). Here we slightly extend the use of this terminology and continue to call it “GFF” when \(p\ne 1\).

  4. 4.

    Throughout the article, C refers to a constant that only depends on the dimension \(\nu \). C’s value may vary from line to line.

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Correspondence to Linan Chen.

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The author is partially supported by NSERC Discovery Grant G241023.

Appendix

Appendix

The Appendix is dedicated to proving Claim 3.9, i.e., the uncorrelation inequality (3.18) between \(\Phi _{x_{j},n}\) and \(\Phi _{x_{k},n}\), which concern random integrals \(X\left( x_{j},t_{n}\right) \) and \(X\left( x_{k},t_{n}\right) \). Let n be large, and take \(x_{j}\) and \(x_{k}\), \(j,k=1,\ldots ,J_{n}\), where \(x_{j}\ne x_{k}\). For convenience, we write \(\delta :=\left| x_{j}-x_{k}\right| \). Assume \(i\in \left\{ 0,1,\ldots ,n-1\right\} \) is the unique integer such that \(2t_{i+1}\le \delta <2t_{i}\) (without loss of generality, we assume that i is large). Let \(\epsilon \) be an arbitrarily small positive constant.

For \(x\in \overline{S\left( O,1\right) }\) and \(0<t<s\le 1\), we denote by \(R\left( x,t\rightarrow s\right) \) the annulus \(B\left( x,s\right) \backslash B\left( x,t\right) \). The increment of the integral \(X\left( x_{j},t\right) -X\left( x_{j},s\right) \) concerns the annulus \(R\left( x_{j},t\rightarrow s\right) \), and similarly, for \(0<t^{\prime }<s^{\prime }\le 1\), \(X\left( x_{k},t^{\prime }\right) -X\left( x_{k},s^{\prime }\right) \) concerns \(R\left( x_{k},t^{\prime }\rightarrow s^{\prime }\right) \). Due to the properties of the covariance function of the family \(\left\{ X\left( x,t\right) :\left( x,t\right) \in \overline{S\left( O,1\right) }\times (0,1]\right\} \), particularly (2.8) and (2.9), we know that \(X\left( x_{j},t\right) -X\left( x_{j},s\right) \) and \(X\left( x_{k},t^{\prime }\right) -X\left( x_{k},s^{\prime }\right) \) are independent if the two corresponding annuli do not intersect, which will happen if either \(B\left( x_{j},s\right) \cap B\left( x_{k},s^{\prime }\right) =\emptyset \) or \(B\left( x_{j},s\right) \subseteq B\left( x_{k},t^{\prime }\right) \) (or \(B\left( x_{k},s^{\prime }\right) \subseteq B\left( x_{j},t\right) \)). This implies that when i is large,

$$\begin{aligned}&\Delta X_{l}\left( x_{j}\right) \text { with }l\in \left\{ i+2,\ldots ,n\right\} ,\,\Delta X_{l^{\prime }}\left( x_{k}\right) \text { with }l^{\prime }\nonumber \\&\in \left\{ 1,\ldots ,i-1,i+2,\ldots ,n\right\} \nonumber \\ \end{aligned}$$
(5.1)

are mutually independent.

To further “extract” independent relationships from the rest of the random variables, we need to look at annuli that intersect and carry out a more careful analysis. Our strategy is to divide the intersected annuli into refined sub-annuli. First, depending on the range of \(\delta \), we will disregard certain sections of the annuli so that the remaining part of the annuli become non-intersecting, which leads to independence of the corresponding integrals. Second, we will verify that removing those sections will only cause a negligible discrepancy in computing the probability of the concerned event, thanks to (a) in Definition 3.1.

We will treat the following two cases separately according to the range of \(\delta \).

Case 1. Suppose that \(t_{i+1}^{1-\epsilon }\le \delta <2t_{i}\). In this case,

$$\begin{aligned} \delta \gg \delta ^{\frac{1}{1-\epsilon /2}}>t_{i+1}^{\frac{1-\epsilon }{1-\epsilon /2}}\gg t_{i+1}. \end{aligned}$$

\(\Delta X_{i+1}\left( x_{j}\right) \) corresponds to the annulus \(R\left( x_{j},t_{i+1}\rightarrow t_{i}\right) \), from which we remove the outer section \(R\left( x_{j},\delta ^{\frac{1}{1-\epsilon /2}}\rightarrow t_{i}\right) \), and only consider the inner annulus \(R\left( x_{j},t_{i+1}\rightarrow \delta ^{\frac{1}{1-\epsilon /2}}\right) \) and set the corresponding integral to be

$$\begin{aligned} Z:=X\left( x_{j},t_{i+1}\right) -X\left( x_{j},\delta ^{\frac{1}{1-\epsilon /2}}\right) . \end{aligned}$$

As for \(\Delta X_{i}\left( x_{k}\right) \) and \(\Delta X_{i+1}\left( x_{k}\right) \), we remove from the corresponding annulus \(R\left( x_{k},t_{i+1}\rightarrow t_{i-1}\right) \) the middle section \(R\left( x_{j},\delta -\delta ^{\frac{1}{1-\epsilon /2}}\rightarrow \delta +\delta ^{\frac{1}{1-\epsilon /2}}\right) \), and set the integral over the remaining two sections as

$$\begin{aligned} W_{1}:=X\left( x_{k},t_{i+1}\right) -X\left( x_{k},\delta -\delta ^{\frac{1}{1-\epsilon /2}}\right) ,\,W_{2}:=X\left( x_{k},\delta +\delta ^{\frac{1}{1-\epsilon /2}}\right) -X\left( x_{k},t_{i-1}\right) . \end{aligned}$$

Z, \(W_{1}\)and \(W_{2}\), as well as all the relevant annuli are indicated in Fig. 1. Now we have that

$$\begin{aligned}&Z,\,W_{1},\,W_{2},\,\Delta X_{l}\left( x_{j}\right) \text { with }l\in \left\{ i+2,\ldots ,n\right\} ,\,\Delta X_{l^{\prime }}\left( x_{k}\right) \\&\quad \text { with }l^{\prime }\in \left\{ 1,\ldots ,i-1,i+2,\ldots ,n\right\} \end{aligned}$$

are all mutually independent centered Gaussian variables.

Fig. 1
figure1

Case 1: \(t_{i+1}^{1-\epsilon }\le \delta <2t_{i}\)

Next, we turn our attention to the probabilities of the events involving Z, \(W_{1}\) and \(W_{2}\). We know that

$$\begin{aligned} \text {Var}\left( Z\right) =\Sigma \left( t_{i+1}\right) -\Sigma \left( \delta ^{\frac{1}{1-\epsilon /2}}\right) \end{aligned}$$

and

$$\begin{aligned} \text {Var}\left( W_{1}+W_{2}\right) =\Sigma \left( t_{i+1}\right) -\Sigma \left( \delta -\delta ^{\frac{1}{1-\epsilon /2}}\right) +\Sigma \left( \delta +\delta ^{\frac{1}{1-\epsilon /2}}\right) -\Sigma \left( t_{i-1}\right) . \end{aligned}$$

The event \(P_{x_{j},i+1}\) implies the event, denoted by \(E_{1}\), that

$$\begin{aligned} \left| Z-\sqrt{2\nu }\text {Var}\left( Z\right) \right| \le 2\sqrt{\Delta \Sigma _{i+1}}, \end{aligned}$$

and \(P_{x_{k},i+1}\cap P_{x_{k},i}\) implying \(E_{2}\), which is the event that

$$\begin{aligned} \left| W_{1}+W_{2}-\sqrt{2\nu }\text {Var}\left( W_{1}+W_{2}\right) \right| \le 4\sqrt{\Delta \Sigma _{i+1}}. \end{aligned}$$

By basic computations, we get that

$$\begin{aligned} {\mathscr {W}}\left( E_{1}\right)\le & {} \exp \left[ -\nu \text {Var}\left( Z\right) +C\sqrt{\Delta \Sigma _{i+1}}\right] \text { and }{\mathscr {W}}\left( E_{2}\right) \\\le & {} \exp \left[ -\nu \text {Var}\left( W_{1}+W_{2}\right) +C\sqrt{\Delta \Sigma _{i+1}^{f}}\right] . \end{aligned}$$

Combining the estimates above with (3.10) and (3.11), we have that

$$\begin{aligned} \begin{aligned}&\frac{{\mathscr {W}}\left( \Phi _{x_{j},n}\cap \Phi _{x_{k},n}\right) }{{\mathscr {W}}\left( \Phi _{x_{j},n}\right) {\mathscr {W}}\left( \Phi _{x_{k},n}\right) }\\&\quad \le \frac{{\mathscr {W}}\left( E_{1}\cap \left( \bigcap _{l=i+2}^{n}P_{x_{j},l}\right) \cap \left( \bigcap _{l=1}^{i-1}P_{x_{j},l}\right) \cap E_{2}\cap \left( \bigcap _{l=i+2}^{n}P_{x_{j},l}\right) \right) }{{\mathscr {W}}\left( \Phi _{x_{j},n}\right) {\mathscr {W}}\left( \Phi _{x_{k},n}\right) }\\&\quad \le \frac{{\mathscr {W}}\left( E_{1}\right) {\mathscr {W}}\left( E_{2}\right) }{{\mathscr {W}}\left( \Phi _{x_{j},i+1}\right) {\mathscr {W}}\left( P_{x_{k},i}\right) {\mathscr {W}}\left( P_{x_{k},i+1}\right) }\\&\quad \le \exp \left[ \nu \Sigma \left( \delta ^{\frac{1}{1-\epsilon /2}}\right) +\nu \Sigma \left( \delta -\delta ^{\frac{1}{1-\epsilon /2}}\right) -\nu \Sigma \left( \delta +\delta ^{\frac{1}{1-\epsilon /2}}\right) +C\sqrt{i\Sigma \left( t_{i+1}\right) }\right] . \end{aligned} \end{aligned}$$

Assuming i is large, the assumption (a) in Definition 3.1 implies that

$$\begin{aligned} \begin{aligned} \Sigma \left( \delta -\delta ^{\frac{1}{1-\epsilon /2}}\right) -\Sigma \left( \delta +\delta ^{\frac{1}{1-\epsilon /2}}\right)&=\int _{\delta +\delta ^{\frac{1}{1-\epsilon /2}}}^{\delta -\delta ^{\frac{1}{1-\epsilon /2}}}f^{2}\left( s\right) {\mathrm{d}}G\left( s\right) \\&\le C\int _{\delta +\delta ^{\frac{1}{1-\epsilon /2}}}^{\delta -\delta ^{\frac{1}{1-\epsilon /2}}}\frac{\left( -\ln s\right) ^{2\rho _{f}}}{G\left( s\right) }{\mathrm{d}}G\left( s\right) \le C\delta ^{\frac{\epsilon }{4-2\epsilon }} \end{aligned} \end{aligned}$$
(5.2)

which is negligible; while (3.8) and the range of \(\delta \) guarantee that

$$\begin{aligned} \sqrt{i\Sigma \left( t_{i+1}\right) }\le \sqrt{i{\tilde{c}}2^{\left( i+1\right) ^{2}}}\le 2^{\frac{2}{3}i^{2}}=o\left( 2^{i^{2}}\right) =o\left( -\ln \delta \right) . \end{aligned}$$

Therefore, in Case 1, when i is sufficiently large,

$$\begin{aligned} \frac{{\mathscr {W}}\left( \Phi _{x_{j},n}\cap \Phi _{x_{k},n}\right) }{{\mathscr {W}}\left( \Phi _{x_{j},n}\right) {\mathscr {W}}\left( \Phi _{x_{k},n}\right) }\le \exp \left[ \nu \Sigma \left( \delta ^{\frac{1}{1-\epsilon /2}}\right) +o\left( -\ln \delta \right) \right] . \end{aligned}$$
(5.3)

Case 2. Suppose that \(2t_{i+1}\le \delta <t_{i+1}^{1-\epsilon }\). In this case, \(t_{i}\gg \delta +t_{i+1}\) and \(\Delta X_{i}\left( x_{k}\right) \) is independent of the family (5.1). We will take the annulus \(R\left( x_{j},t_{i+2}\rightarrow t_{i+1}\right) \) and remove from it the outer section \(R\left( x_{j},t_{i+1}^{1+\epsilon }\rightarrow t_{i+1}\right) \), and set the integral over the remaining inner section \(R\left( x_{j},t_{i+2}\rightarrow t_{i+1}^{1+\epsilon }\right) \) to be

$$\begin{aligned} Z^{\prime }=X\left( x_{j},t_{i+2}\right) -X\left( x_{j},t_{i+1}^{1+\epsilon }\right) . \end{aligned}$$

We also remove from \(R\left( x_{k},t_{i+1}\rightarrow t_{i}\right) \) the middle section \(R\left( x_{k},\delta -t_{i+1}^{1+\epsilon }\rightarrow \delta +t_{i+1}^{1+\epsilon }\right) \), and set the integral over the remaining two sections as

$$\begin{aligned} W^{\prime }=X\left( x_{k},t_{i+1}\right) -X\left( x_{k},\delta -t_{i+1}^{1+\epsilon }\right) +X\left( x_{k},\delta -t_{i+1}^{1+\epsilon }\right) -X\left( x_{k},t_{i}\right) . \end{aligned}$$

See Fig. 2 for an illustration of the division of the annuli. By doing the above, we have that

$$\begin{aligned}&Z^{\prime },W^{\prime },\Delta X_{l}\left( x_{j}\right) \text { with }l\in \left\{ i+3,\ldots ,n\right\} ,\,\Delta X_{l^{\prime }}\left( x_{k}\right) \text { with }l^{\prime }\\&\quad \in \left\{ 1,\ldots ,i,i+2,\ldots ,n\right\} \end{aligned}$$

are mutually independent. Furthermore,

$$\begin{aligned} \text {Var}\left( Z^{\prime }\right) =\Sigma \left( t_{i+2}\right) -\Sigma \left( t_{i+1}^{1+\epsilon }\right) \end{aligned}$$

and

$$\begin{aligned} \text {Var}\left( W^{\prime }\right) =\Sigma \left( t_{i+1}\right) -\Sigma \left( \delta -t_{i+1}^{1+\epsilon }\right) +\Sigma \left( \delta +t_{i+1}^{1+\epsilon }\right) -\Sigma \left( t_{i}\right) . \end{aligned}$$
Fig. 2
figure2

Case 2: \(t_{i+1}^{1-\epsilon }\le \delta <2t_{i}\)

Similarly as in the previous case, we have that

$$\begin{aligned} P_{x_{j},i+2}\subseteq E_{1}^{\prime }:= & {} \left\{ \left| Z^{\prime }-\sqrt{2\nu }\text {Var}\left( Z^{\prime }\right) \right| \le 2\sqrt{\Delta \Sigma _{i+2}}\right\} ,\;{\mathscr {W}}\left( E_{1}^{\prime }\right) \\\le & {} \exp \left( -\nu \text {Var}\left( Z^{\prime }\right) +C\sqrt{\Delta \Sigma _{i+2}}\right) , \end{aligned}$$

and

$$\begin{aligned}&P_{x_{k},i+1}\subseteq E_{2}^{\prime }:=\left\{ \left| W^{\prime }-\sqrt{2\nu }\text {Var}\left( W\right) \right| \le 3\sqrt{\Delta \Sigma _{i+1}}\right\} ,\;\\&{\mathscr {W}}\left( E_{2}^{\prime }\right) \le \exp \left( -\nu \text {Var}\left( W^{\prime }\right) +C\sqrt{\Delta \Sigma _{i+1}}\right) . \end{aligned}$$

Following the same arguments as in (5.2), we get that

$$\begin{aligned} \begin{aligned} \Sigma \left( \delta -t_{i+1}^{1+\epsilon }\right) -\Sigma \left( \delta +t_{i+1}^{1+\epsilon }\right)&=\int _{\delta +t_{i+1}^{1+\epsilon }}^{\delta -t_{i+1}^{1+\epsilon }}f^{2}\left( s\right) {\mathrm{d}}G\left( s\right) \\&\le C\int _{\delta +t_{i+1}^{1+\epsilon }}^{\delta -t_{i+1}^{1+\epsilon }}\frac{\left( -\ln s\right) ^{2\rho _{f}}}{G\left( s\right) }{\mathrm{d}}G\left( s\right) \le Ct_{i+1}^{\epsilon /4}. \end{aligned} \end{aligned}$$

Combining the arguments above with (3.10) and (3.11) leads to

$$\begin{aligned} \begin{aligned}&\frac{{\mathscr {W}}\left( \Phi _{x_{j},n}\cap \Phi _{x_{k},n}\right) }{{\mathscr {W}}\left( \Phi _{x_{j},n}\right) {\mathscr {W}}\left( \Phi _{x_{k},n}\right) } \\&\quad \le \frac{{\mathscr {W}}\left( E_{1}^{\prime }\cap \left( \bigcap _{l=i+3}^{n}P_{x_{j},l}\right) \cap \left( \bigcap _{l^{\prime }=1}^{i}P_{x_{k},l^{\prime }}\right) \cap E_{2}^{\prime }\cap \left( \bigcap _{l^{\prime }=i+2}^{n}P_{x_{k},l^{\prime }}\right) \right) }{{\mathscr {W}}\left( \Phi _{x_{j},n}\right) {\mathscr {W}}\left( \Phi _{x_{k},n}\right) }\\&\quad \le \frac{{\mathscr {W}}\left( E_{1}^{\prime }\right) {\mathscr {W}}\left( E_{2}^{\prime }\right) }{{\mathscr {W}}\left( \Phi _{x_{j},i+2}\right) {\mathscr {W}}\left( P_{x_{k},i+1}\right) }\le \exp \left[ \nu \Sigma \left( t_{i+1}^{1+\epsilon }\right) +C\sqrt{i\Sigma \left( t_{i+2}\right) }\right] . \end{aligned} \end{aligned}$$

When i is large, again by (3.8) and the range of \(\delta \), we have that \(\Sigma \left( t_{i+1}^{1+\epsilon }\right) \le \Sigma \left( \delta ^{\frac{1+\epsilon }{1-\epsilon }}\right) \) and \(\sqrt{i\Sigma \left( t_{i+2}\right) }=o\left( -\ln \delta \right) \). In other words, in Case 2 we have that

$$\begin{aligned} \frac{{\mathscr {W}}\left( \Phi _{x_{j},n}\cap \Phi _{x_{k},n}\right) }{{\mathscr {W}}\left( \Phi _{x_{j},n}\right) {\mathscr {W}}\left( \Phi _{x_{k},n}\right) }\le \exp \left[ \nu \Sigma \left( \delta ^{\frac{1+\epsilon }{1-\epsilon }}\right) +o\left( -\ln \delta \right) \right] . \end{aligned}$$
(5.4)

Combining (5.3) and (5.4) together, since \(\frac{1+\epsilon }{1-\epsilon }>\frac{1}{1-\epsilon /2}\), we have arrived at (3.18).

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Chen, L. Steep Points of Gaussian Free Fields in Any Dimension. J Theor Probab (2020). https://doi.org/10.1007/s10959-020-01028-7

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Keywords

  • Gaussian free field
  • Regularization processes
  • Steep point
  • Thick point
  • Exceptional set
  • Hausdorff dimension

Mathematics Subject Classification 2010

  • 60G60
  • 60G15