Related papers: Maximum of the integer-valued Gaussian free field
We show that if an interlacing particle system in a two-dimensional lattice is a determinantal point process, and the correlation kernel can be expressed as a double integral with certain technical assumptions, then the moments of the…
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…
We study the laws of the two-dimensional vector-valued Dirichlet Gaussian free field and its massive lattice counterpart, conditioned to avoid a ball at every site of a subdomain. We prove that, under this conditioning, the norm of the…
We consider a class of Gaussian Free Fields denoted by $(g_x)_{x \in {\cal V}_N}$, where $ {\cal V}_N = \{0,1\}^N$ and $N\in \mathbb{Z}_+$. These fields are related to a general class of $N$-dimensional random walks on the hypercube, which…
These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text…
We review the construction of free gauge theories for gauge fields in arbitrary representations of the Lorentz group in $D$ dimensions. We describe the multi-form calculus which gives the natural geometric framework for these theories. We…
We consider the discrete Ginzburg-Landau field with potential satisfying a uniform convexity condition, in the critical dimension $d=2$, and prove that its maximum over boxes of sidelength $N$, centered by an explicit $N$-dependent…
The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields in one variable over a henselian valued field with…
We compute the leading asymptotics as $N\to\infty$ of the maximum of the field $Q_N(q)= \log\det|q- A_N|$, $q\in \mathbb{C}$, for any unitarily invariant Hermitian random matrix $A_N$ associated to a non-critical real-analytic potential.…
Let $\alpha(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $\alpha(\mathbb{F}_q^{d},p)$ up to a polylogarithmic factor…
The classical scalar massive field satisfying the Klein-Gordon equation in a finite one-dimensional space interval of periodically varying length with Dirichlet boundary conditions is studied. For the sufficiently small mass, the energy can…
In [HLS], N. Hindman, I. Leader and D. Strauss proved the abundance for a matrix with rational entries. In this paper we proved it for the ring of Gaussian integers. We showed the result when the matrix is taken with entries from…
The assumption that the elements of the cost matrix in the classical assignment problem are drawn independently from a standard Gaussian distribution motivates the study of a particular Gaussian field indexed by the symmetric permutation…
We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in…
Lagrangian formulation of free massive fields corresponding to irreducible representations of the Poincare group of arbitrary integer and half-integer spins in three-dimensional space-time is presented. A relationship of the theory under…
For a family of integer-valued height functions defined over the faces of planar graphs, we establish a relation between the probability of connection by level sets and the spin-spin correlations of the dual $O(2)$ symmetric spin models…
In this paper, we consider the maximum of the $\text{Sine}_\beta$ counting process from its expectation. We show the leading order behavior is consistent with the predictions of log-correlated Gaussian fields, also consistent with work on…
We study the lattice of T-spaces of a free associative k-algebra over a nonempty set. It is shown that when the field k is infinite, then the lattice has a maximum element, and that maximum element is in fact a T-ideal. In striking…
We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is…
Massive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal…