Discrete Gaussian Free Field Research Articles

Near-maxima of the two-dimensional discrete Gaussian free field

Near-maxima of the two-dimensional discrete Gaussian free field

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Finite range decompositions of Gaussian fields with applications to level-set percolation

In a recent work [19], Muirhead has studied level-set percolation of (discrete or continuous) Gaussian fields, and has shown sharpness of the associated phase transition under the assumption that the field has a certain multiscale white noise decomposition, a variant of a finite-range decomposition. We show that a large class of Gaussian fields have such a white noise decomposition with optimal decay parameter. Examples include the discrete Gaussian free field, the discrete membrane model, and the mollified continuous Gaussian free field. This answers various questions from [19].Our construction of the white-noise decomposition is a refinement of Bauerschmidt's construction of a finite-range decomposition [3]. In the continuous setting our construction is very similar to Bauerschmidt's, while in the discrete setting several new ideas are needed, including the use of a result by Pólya and Szegő on polynomials that take positive values on the positive real line.

Properties of the Gradient Squared of the Discrete Gaussian Free Field

In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in Uε=U/ε∩Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_{\\varepsilon }=U/\\varepsilon \\cap \\mathbb {Z}^d$$\\end{document}, U⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\subset \\mathbb {R}^d$$\\end{document} and d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}. The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-Hölder space. Moreover, under a different rescaling, we determine the k-point correlation function and joint cumulants on Uε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_{\\varepsilon }$$\\end{document} and in the continuum limit as ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document}. This result is related to the analogue limit for the height-one field of the Abelian sandpile (Dürre in Stoch Process Appl 119(9):2725–2743, 2009), with the same conformally covariant property in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}.

A branching random walk in the presence of a hard wall

AbstractWe consider a branching random walk on a d-ary tree of height n ( $n \in \mathbb{N}$ ), in the presence of a hard wall which restricts each value to be positive, where d is a natural number satisfying $d\geqslant2$ . We consider the behaviour of Gaussian processes with long-range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of vertices. We observe a relation with the expected maximum of the processes. We find the probability of the event that the branching random walk is positive at every vertex in the nth generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$ .

Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333–380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395–1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771–833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191–225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087–1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions, stays negative as the ratio of the radii of the outer and the inner boundary tends to infinity. Such estimates are often needed in the study of extreme values of the discrete Gaussian free field on planar domains.

Exceptional points of two-dimensional random walks at multiples of the cover time

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by N) versions $$D_N\subseteq {\mathbb {Z}}^2$$ of bounded open domains $$D\subseteq {\mathbb {R}}^2$$ . Upon exit from $$D_N$$ , the walk lands on a “boundary vertex” and then reenters $$D_N$$ through a random boundary edge in the next step. In the parametrization by the local time at the “boundary vertex” we prove that, at times corresponding to a $$\theta $$ -multiple of the cover time of $$D_N$$ , the sets of suitably defined $$\lambda $$ -thick (i.e., heavily visited) and $$\lambda $$ -thin (i.e., lightly visited) points are, as $$N\rightarrow \infty $$ , distributed according to the Liouville Quantum Gravity $$Z^D_\lambda $$ with parameter $$\lambda $$ -times the critical value. For $$\theta <1$$ , also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to $$Z^D_{\sqrt{\theta }}$$ . The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.

Pinning for the critical and supercritical membrane model

The membrane model is a Gaussian interface model with a Hamiltonian involving second derivatives of the interface height. We consider the model in dimension $\mathsf{d}\ge4$ under the influence of $\delta$-pinning of strength $\varepsilon$. It is known that this pinning potential manages to localize the interface for any $\varepsilon>0$. We refine this result by establishing the $\varepsilon$-dependence of the variance and of the exponential decay rate of the covariances for small $\varepsilon$ (similar to the corresponding results for the discrete Gaussian free field by Bolthausen-Velenik). We also show the existence of a thermodynamic limit of the field. These conclusions improve upon earlier works by Bolthausen-Cipriani-Kurt and by Sakagawa. The problem has similarities to the homogenization of elliptic operators in randomly perforated domains, and our proof takes inspiration from this connection. The main new ideas are a correlation inequality for the set of pinned points, and a probabilistic Widman hole filler argument which relies on a discrete multipolar Hardy-Rellich inequality and on a multi-scale argument to construct suitable test functions.

The distance exponent for Liouville first passage percolation is positive

Discrete Liouville first passage percolation (LFPP) with parameter \(\xi > 0\) is the random metric on a sub-graph of \(\mathbb Z^2\) obtained by assigning each vertex z a weight of \(e^{\xi h(z)}\), where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all \(\xi > 0\). More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size \(2^n\) is typically at least \(2^{\alpha n}\) for an exponent \(\alpha > 0\) depending on \(\xi \). This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all \(\xi > 0\) and also has theoretical implications for the study of distances in Liouville quantum gravity.

Covariant Symanzik identities

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject.

A scaling limit for the cover time of the binary tree

We consider a continuous time random walk on the rooted binary tree of depth n with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by 2n+1n and then centered by (log⁡2)n−log⁡n, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree.

Two-temperatures overlap distribution for the 2D discrete Gaussian free field

In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM’s one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Extremal process in the weakly correlated regime

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Extremal process in the weakly correlated regime

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Convergence of the maximum in the regime of weak correlations

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Convergence of the maximum in the regime of weak correlations

Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection

We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the occupation-time profile undergoes an entropic push governed by a specific function depending on $A$. Similar entropic repulsion phenomena conditioned on disconnection by level-sets of the discrete Gaussian free field on $\mathbb{Z}^d$, $d \geq 3$, have been obtained by the authors in arxiv:1808.09947. Our proofs rely crucially on the `solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.

Conformal Symmetries in the Extremal Process of Two-Dimensional Discrete Gaussian Free Field

We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions (x) and centered values (h) of the extreme local maxima in lattice versions of a bounded domain $$D\subset {\mathbb {C}}$$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $$Z^D(\mathrm{d}x)\otimes \mathrm{e}^{-\alpha h}\mathrm{d}h$$, where $$\alpha $$ is a constant and $$Z^D$$ is a random a.s.-finite measure on D. The random measures $$\{Z^D\}$$ are naturally interrelated; restrictions to subdomains are governed by a Gibbs–Markov property and images under analytic bijections f by the transformation rule $$(Z^{f(D)}\circ f)(\mathrm{d}x)\,\overset{\mathrm{law}}{=}\,|f'(x)|^4\, Z^D(\mathrm{d}x)$$. Conditions are given that determine the laws of these measures uniquely. These identify $$Z^D$$ with the critical Liouville Quantum Gravity associated with the Continuum Gaussian Free Field.

An Analysis of the Induced Linear Operators Associated to Divide and Color Models

We study the natural linear operators associated to divide and color (DC) models. The degree of nonuniqueness of the random partition yielding a DC model is directly related to the dimension of the kernel of these linear operators. We determine exactly the dimension of these kernels as well as analyze a permutation-invariant version. We also obtain properties of the solution set for certain parameter values which will be important in (1) showing that large threshold discrete Gaussian free fields are DC models and in (2) analyzing when the Ising model with a positive external field is a DC model, both in future work. However, even here, we give an application to the Ising model on a triangle.

Maximum of the Ginzburg–Landau fields

We study a two-dimensional massless field in a box with potential $V(\nabla \phi (\cdot ))$ and zero boundary condition, where $V$ is any symmetric and uniformly convex function. Naddaf–Spencer (Comm. Math. Phys. 183 (1997) 55–84) and Miller (Comm. Math. Phys. 308 (2011) 591–639) proved that the rescaled macroscopic averages of this field converge to a continuum Gaussian free field. In this paper, we prove that the distribution of local marginal $\phi (x)$, for any $x$ in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and the dimension of high points of this field, thus generalizing the results of Bolthausen–Deuschel–Giacomin (Ann. Probab. 29 (2001) 1670–1692) and Daviaud (Ann. Probab. 34 (2006) 962–986) for the discrete Gaussian free field.

Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

We investigate level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level $\alpha$ disconnects the discrete blow-up of a compact set $A$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of $A$, when disconnection occurs. These bounds, combined with the findings of the recent article [12], show that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of $A$. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain 'profile' description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of $A$, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the 'solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.