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The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

Abstract

We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used. In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the Gaussian free field along this path. We first derive a pathwise transformation that relates the GFF on the metric graph with the reflected GFF on the metric graph via the pseudo-distance defined by the latter. This is a generalisation of Paul Lévy’s result relating the local time at zero of Brownian motion to the supremum of another Brownian motion. We also compute explicitly the distribution of certain functionals of this pseudo-metric and of the GFF. In particular, we point out that when the boundary consists of just two points, the law of the pseudo-distance between them depends solely on the resistance of the network between them. We then discuss questions related to the scaling limit of this pseudo-metric in the two-dimensional case, which should be the conformally invariant way to measure distances between CLE(4) loops introduced and studied by the second author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum GFF on fairly general Riemann surfaces.

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Notes

  1. 1.

    Since this formula will be used throughout the paper, let us recall briefly a possible direct proof: For a Brownian motion W started from \(W_0\), one knows from the reflection principle [28] the joint law of \((W_T, I_T)\), from which one can deduce the law of \((|W_T|, L_T)\) by Lévy’s theorem. We note that by reflection, the laws of \((W_T, L_T) 1_{L_T >0}\) and of \((-W_T, L_T) 1_{L_T >0}\) coincide while \(L_T = 0\) implies that \(W_T\) has the same sign as \(W_0\). From this, one can deduce the joint law of \((W_T, L_T)\), and by conditioning on the value of \(W_T\), one gets the law of \(L_T\) for the Brownian bridge.

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Acknowledgements

T.L. acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. W.W. acknowledges the support of the SNF Grant SNF-155922. The authors are also part of the NCCR Swissmap of the SNF. They also thank the referees for their comments.

Author information

Correspondence to Titus Lupu.

Additional information

Dedicated to the memory of Marc Yor.

Appendix

Appendix

Here we provide some details about the rather brute-force Itô formula computation of the semi-martingale decomposition of \(\varPsi _{\ell }(t)\) that proves Lemma 8. We use the notation of the beginning of Sect. 3. To simplify the expressions, we will use the following notation:

At each time t, the \(\tilde{\phi }^{*}_{i}(t)\) for different values of i have the same sign. We will denote this common sign by \(\sigma (t)\in \lbrace -1,+1\rbrace \). If all the \(\tilde{\phi }^{*}_{i}(t)\) are zero, the value of \(\sigma (t)\) does not matter. Similarly, we will denote the common sign of h on . If h vanishes on , the value of the sign does not matter.

According to Lemma 2, \(\tilde{\phi }^{*}_{i}(t)\) has the following semi-martingale decomposition:

where \(M^{*}_{i}(t)\) is an \((\widehat{{\mathcal {F}}}_{t})_{t\geqslant 0}\)-martingale. Moreover, \(\langle M_{i}^{*},M_{i}^{*}\rangle _{t}=r^{*}_{i}(t)\) and \( \langle M_{i}^{*},M_{j}^{*}\rangle _{t}=0\) for \(i\ne j\). Further, \(d S_{i\check{x}}(t)= d \tilde{\phi }^{*}_{i}(t)\), \(d S_{i\check{x}}^{+}(t)= \sigma (t) d\tilde{\phi }^{*}_{i}(t)\), and \(d\langle S_{i\check{x}},S_{i\check{x}}\rangle _{t}= d\langle S^{+}_{i\check{x}},S^{+}_{i\check{x}}\rangle _{t} = d r^{*}_{i}(t).\) According to Lemma 3, Eqs. (8) and (9),

Using Itô’s formula we get that

where \(\theta _{\check{x}}(t)=(\sigma (t)\vert h(\check{x})\vert + h(\check{x})+\sigma (t)(\ell -L^{*}(t)))\). Further

Finally,

It follows that \(\varPsi _{\ell }(t)\) is a local martingale.

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Lupu, T., Werner, W. The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph. Probab. Theory Relat. Fields 171, 775–818 (2018). https://doi.org/10.1007/s00440-017-0792-y

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Keywords

  • Random distances
  • Gaussian free field
  • Metric graph
  • Local time
  • Conformal loop ensemble

Mathematics Subject Classification

  • 60G60
  • 60G15
  • 60K35
  • 60J55
  • 60J65
  • 60J67