v2: Updated to take into account the subsequent paper by D. Hua, S. Tudose and myself arXiv:2605.10908, which solves some of the main questions following the approach of this paper. Added comments explaining what is (not) subsumed by arXiv:2605.10908. Proposed a Riemannian version of the convexity problem of M. Talagrand
We prove that the convexity problem of M. Talagrand is equivalent to the subgaussian vector problem: can any centered $1$-subgaussian random vector in $\mathbb{R}^n$ be realized as the sum of a universal number of standard Gaussian vectors? We introduce methods to study this problem and, using elementary arguments, we settle it for $1$-subgaussian random variables and random vectors with good norm and covariance bounds. These results already confirm the permutation invariant case of the convexity problem, and give optimal estimates on the largest ellipsoid contained in a sum of large sets in Gaussian spaces. We also propose a Riemannian version of the convexity problem for spaces with nonnegative Ricci curvature.
Given $2\le k\le n$, the minimal $(n-1)$-dimensional Gaussian measure of the union of the boundaries of $k$ disjoint sets of equal Gaussian measure in $\R^n$ whose union is $\R^n$ is of order $\sqrt{\log k}$. A similar results holds also for partitions of the sphere $S^{n-1}$ into $k$ sets of equal Haar measure.
A sharp Poincaré-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure.
The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. In this paper we obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, c, such that the conjecture is true if the two sets are in the Euclidean ball of radius $c\sqrt{n}$. Further we show that if for every n the conjecture is true when the sets are in the Euclidea…
This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.
Gaussian correlation conjecture states that the Gaussian measure of the intersection of two symmetric convex sets is greater or equal to the product of the measures.
In this short note, we present a theorem concerning certain "additive structure" for the level sets of non-degenerate Gaussian fields, which yields the multiple valley phenomenon for extremal fields with exponentially many valleys.
The paper is to prove the Gaussian correlation conjecture stating that, under the standard Gaussian measure, the measure of the intersection of any two symmetric convex sets is greater than or equal to the product of their measures. Characterization of the equality and some applications are given.
Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times. Most recently a CLT for Euler-Poincaré characteristic of the excursions set of a Gaussian random field has been proven under appropriate conditions. In this paper, we shall obtain a central limit theorem for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriat…
We prove decoupling inequalities for the Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$. As an application, we obtain exponential decay (with logarithmic correction for $d=3$) of the connectivity function of excursion sets for large values of the threshold.
We consider the complex case of the so-called S-inequality. It concerns the behaviour of the Gaussian measures of dilations of convex and rotationally symmetric sets in C^n (rotational symmetry is invariance under the multiplication by $e^{it}$, for any real t). We pose and discuss a conjecture that among all such sets the measure of cylinders (i.e. the sets $\{z \in C^n: |z_1| \leq p\}$) decrease the fastest under dilations. Our main result of the paper is that this conjecture holds under the additional assumption that the Gaussian measure of considered sets is not greater than some constant …
This paper studies polar sets of anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random f…
This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-Itô integrals. We illustrate most results with corresponding examples.
We consider a subordinate Brownian motion $X$ with Gaussian components when the scaling order of purely discontinuous part is between $0$ and $2$ including $2$. In this paper we establish sharp two-sided bounds for transition density of $X$ in ${\mathbb R}^d$ and $C^{1,1}$-open sets. As a corollary, we obtain a sharp Green function estimates.
We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $γ_d(A\cap B)\geq γ_d(A)γ_d(B)$, where $γ_d$ is the standard Gaussian measure on $\mathbb{R}^d$. This generalizes Proposition 1 in [Arch. Rational Mech. Anal. 161 (2002), 257--269].
We study the Gaussian noise stability of subsets A of Euclidean space satisfying A=-A. It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the Symmetric Gaussian Problem.
We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.
We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.
We study the probability distribution of the maximum $M_S $ of a smooth stationary Gaussian field defined on a fractal subset $S$ of $\R^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $¶(M_S>u)$ as $u\rightarrow +\infty.$ The basic tool is Rice formula for the moments of the number of local maxima of a random field.
We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.
Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and investigate some of its properties.
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations. The special case $m=3$ proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when $m=3$, the three minimal sets are adjacent $120$ degree sectors. The technical assumption is that the triple junction points of the minimizing sets have polynomial volume growth. Assuming again the technical assumption, we prove …
We study deviation probabilities for the number of high positioned particles in branching Brownian motion, and confirm a conjecture of Derrida and Shi (2016). We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton-Watson processes.
We prove Fatou's theorem for nonnegative harmonic functions with respect to subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential conver…
The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.
We study how small a local set of the continuum Gaussian free field (GFF) in dimension $d$ has to be to ensure that this set is thin, which loosely speaking means that it captures no GFF mass on itself, in other words, that the field restricted to it is zero. We provide a criterion on the size of the local set for this to happen, and on the other hand, we show that this criterion is sharp by constructing small local sets that are not thin.
We consider the Gaussian free field on $\mathbb{Z}^d$, $d$ greater or equal to $3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1 + o(1)}$ as $d$ tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level $h_*(d)$. Moreover, it shows that a related parameter $h_{**}(d) \geq h_*(d)$ introduced by Rodriguez and Sznitman in arXiv:1202.5172 is in fact asymptotically equivalent to $h_*(d)$.
When a random field $(X_t, \ t\in {\mathbb R}^2)$ is thresholded on a given level $u$, the excursion set is given by its indicator $~1_{[u, \infty)}(X_t)$. The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets, as e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular Rice methods, and from integral and stochastic geometry.
In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Aza\"ıs and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Aza\"ıs and Wschebor that shows that under some conditi…
We continue the research of Latała on improving estimates of $p$-th moments of sums of independent random variables. We generalize some of his results in the case when $2 \leq p \leq 4$ and present a combinatorial approach for even moments.
Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic compact manifolds. Our work is motivated by questions about the geometry of such random functions, in particular relating to the structure of their nodal and level sets. We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to a different level of precision, and each req…
The $r$-parallel set of a measurable set $A \subseteq \mathbb R^d$ is the set of all points whose distance from $A$ is at most $r$. In this paper, we show that the surface area of an $r$-parallel set in $\mathbb R^d$ with volume at most $V$ is upper-bounded by $e^{Θ(d)}V/r$, whereas its Gaussian surface area is upper-bounded by $\max(e^{Θ(d)}, e^{Θ(d)}/r)$. We also derive a reverse form of the Brunn-Minkowski inequality for $r$-parallel sets, and as an aside a reverse entropy power inequality for Gaussian-smoothed random variables. We apply our results to two problems in theoretical machine le…
We study level-set percolation for Gaussian free fields on metric graphs. In two dimensions, we give an upper bound on the chemical distance between the two boundaries of a macroscopic annulus. Our bound holds with high probability conditioned on connectivity and the bound is sharp up to a poly-logarithmic factor with an exponent of one-quarter. This substantially improves a previous result by Li and the first author. In three dimensions and higher, we provide rather sharp estimates of percolation probabilities in different regimes which altogether describe a sharp phase transition.
Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussia…
The Nazarov-Sodin constant describes the average number of nodal set components of Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important s…
Let $\mathcal{T}$ be a rooted tree endowed with the natural partial order $\preceq$. Let $(Z(v))_{v\in \mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $α= (α_k)_{k=1}^\infty$ be a sequence of real numbers with $\sum_{k=1}^\infty α_k^2<\infty$. Set $α_0 =0$ and define a Gaussian process on $\mathcal{T}$ in the following way: \[ G(\mathcal{T}, α; v): = \sum_{u\preceq v} α_{|u|} Z(u), \quad v \in \mathcal{T}, \] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $\mathcal{T}$, we obtain a necessary and…
We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that $|λ| \leq 2\sqrt{d-1}$. We then use these approximations for $λ= -2\sqrt{d-1}$ to produce factor of i.i.d. ind…
In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane. We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kern…
We provide new constructions of the subcritical and critical Gaussian multiplicative chaos (GMC) measures corresponding to the 2D Gaussian free field (GFF). As a special case we recover E. Aidekon's construction of random measures using nested conformally invariant loop ensembles, and thereby prove his conjecture that certain CLE$_4$ based limiting measures are equal in law to the GMC measures for the GFF. The constructions are based on the theory of local sets of the GFF and build a strong link between multiplicative cascades and GMC measures. This link allows us to directly adapt techniques …
The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivations arising from Physics and Cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a Uniform Central Limit Theorem for the empirical measure, i.e. the proportion of spherical surface where spherical Gaussian eigenfunctions lie below a level $z$. Our proofs borrows some techniques from the literature on stationary long memory processes; in particular, we expand…
We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply-connected domain, and their relation to the conformal loop ensemble CLE(4) and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded-type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the CLE(4) carpet, and prove that all…
We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors ($p$) is large compared to the sample size ($n$); in fact, $p$ can be much larger than $n$, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum…
