Gaussian fields on a hypercube from long range random walks

Gaussian multiplicative chaos is a random measure constructed from a Gaussian field. An example of this is the Liouville measure, which is constructed from a Gaussian free field. Under certain technical assumptions, we prove the convergence of a process time-changed by Gaussian multiplicative chaos in the case the latter object is square integrable (the $$L^2$$ L 2 -regime). As examples of the main result, we prove that, in the whole $$L^2$$ L 2 -regime, the scaling limit of the Liouville simple random walk on $$\mathbb {Z}^2$$ Z 2 is Liouville Brownian motion and, as $$\alpha \rightarrow 1$$ α → 1 , Liouville $$\alpha $$ α -stable processes on $$\mathbb {R}$$ R converge weakly to the Liouville Cauchy process.

It has been recently understood [9, 24, 30] that for a general class of percolation models on $\mathbb{Z} ^{d}$ satisfying suitable decoupling inequalities, which includes i.a. Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT. In this paper we consider the random walk loop soup on $\mathbb{Z} ^{d}$ in dimensions $d\geq 3$. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since the required decoupling inequalities are not valid. We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang [6] by a model specific approximation method, which does not apply to the vacant set.) Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$ ($d\geq 2$). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and {the vacant set of random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free field ($d\geq 3$). Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the {\it{pair empirical measures}} of the environment Markov chain in the non-elliptic case of SRWPC . Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main difficulty in our set up lies in the inherent non-ellipticity as well as the lack of {\it{translation-invariance}} stemming from conditioning on the fact that the origin belongs to the infinite cluster. We develop a unifying approach for proving quenched large deviations for SRWPC based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster.

We characterise the multiplicative chaos measure {mathcal {M}} associated to planar Brownian motion introduced in Bass et al. (Ann Probab 22(2):566–625, 1994), Aïdékon et al. (Ann. Probab. 48(4), 1785–1825, 2020) and Jego (Ann Probab 48(4):1597–1643, 2020) by showing that it is the only random Borel measure satisfying a list of natural properties. These properties only serve to fix the average value of the measure and to express a spatial Markov property. As a consequence of our characterisation, we establish the scaling limit of the set of thick points of planar simple random walk, stopped at the first exit time of a domain, by showing the weak convergence towards {mathcal {M}} of the point measure associated to the thick points. In particular, we obtain the convergence of the appropriately normalised number of thick points of random walk to a nondegenerate random variable. The normalising constant is different from that of the Gaussian free field, as conjectured in Jego (Electron J Probab 25:39, 2020). These results cover the entire subcritical regime. A key new idea for this characterisation is to introduce measures describing the intersection between different independent Brownian trajectories and how they interact to create thick points.

We consider the scaling behavior of the range and p-multiple range, that is the number of points visited and the number of points visited exactly p≥1 times, of a simple random walk on ℤd, for dimensions d≥2, up to time of exit from a domain DN of the form DN=ND, where D⊂ℝd, as N↑∞. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case D is a cube in d≥3, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in d≥2, both weakly converge to proportional exit times of Brownian motion from D, and that the corresponding limit moments are “polyharmonic”, solving a hierarchy of Poisson equations.

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm–Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, “large” lattice loops from the random walk loop soup converge to “large” loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of “large” lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of “large” lattice loops converges to CLE.

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\gamma X(z)}\,dz^{2}$, $\gamma<\gamma_{c}=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_{t}$ depending on the local behavior of the Liouville measure “$M_{\gamma}(dz)=e^{\gamma X(z)}\,dz$”. We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_{c}=2$ and that for all $\gamma<\gamma_{c}$, the Liouville measure $M_{\gamma}$ is invariant under $P_{\mathbf{t}}$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.

Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field

This paper investigates wrapped and unwrapped phase differences generated by a non-Gaussian scattering model: the two-dimensional random walk. Mean square values for these quantities are obtained for one and two scatterers, as well as the large scatterer limit when the field constitutes a circular complex Gaussian process. Numerical simulation is used to investigate the phase under more general fluctuation conditions, and reveals that the wrapped phase difference correlation converges rapidly to that result predicted for a Gaussian speckle field. Analytical results for the unwrapped phase indicate that this quantity transitions from a stationary process for one and two scatterers to a non-stationary process in the large scatterer limit. The nature of this transition is examined using numerical simulation for arbitrary scatterer number. Phase correlations are of consequence in various phase sensitive detection systems, and this paper examines both Gaussian and non-Gaussian fields.

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.

The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm–Loewner evolution.

Fractional Brownian Motions, Fractional Noises and Applications

The loop-erased random walk (LERW) in $\mathbb{Z}^{4}$ is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost surely and in $L^{p}$ for all $p>0$. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a $\pm 1$ spin model coupled with the wired spanning forests on $\mathbb{Z}^{4}$ with the bi-Laplacian Gaussian field on $\mathbb{R}^{4}$ as its scaling limit.

The goal of this work is to study the uncertainty quantification in inverseproblems of miscible flows in heterogeneous porous media in Bayesian framework and propose a permeability update based on observed measurements of spatially sparse tracer concentration at certain times. The Successive Sum of Independent Gaussian Fields method is used to parametrization of the uncertainty in heterogeneous porous media. A two-stageMarkov chain Monte Carlo (MCMC) method is used to solving the inverse problem. For the construction of Markov chains is used the Metropolis-Hastings algorithm based on a random walk sampling process employs an auto-regressive instrumental distribution. The main idea of MCMC is to generate a Markov chain with an equilibrium distribution equal to the posterior distribution of interest. Numerical results are presented for a set sampled realizations of the permeability fields.