We consider upper level sets of the Gaussian free field (GFF) on Zd, for d≥3, above a given real-valued height parameter h. As h varies, this defines canonical percolation model with strong, algebraically decaying correlations. prove that three natural critical parameters associated model, respectively describing well-ordered subcritical phase, emergence an infinite cluster, and onset local uniqueness regime in supercritical actually coincide. At core our proof lies new interpolation scheme...

Abstract For a large class of amenable transient weighted graphs G , we prove that the sign clusters Gaussian free field on fall into regime strong supercriticality in which two infinite dominate (one for each sign), and finite are necessarily tiny, with overwhelming probability. Examples belonging to this include regular lattices such as $${\mathbb {Z}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi>...

We consider the bond percolation problem on a transient weighted graph induced by excursion sets of Gaussian free field corresponding cable system. Owing to continuity this setup and strong Markov property one hand, links with potential theory for associated diffusion other, we rigorously determine behavior various key quantities related (near-)critical regime model. In particular, our results apply in case base is three-dimensional cubic lattice. They unveil values critical exponents, which...

We consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$, for $d\geq3$, and give sharp bounds probability that radius of a finite cluster in excursion set $\{\varphi \geq h\}$ exceeds large value $N$, any height $h \neq h_*$, where $h_*$ refers to corresponding percolation critical parameter. In dimension $d=3$, we prove this is sub-exponential $N$ decays as $\exp\{-\frac{\pi}{6}(h-h_*)^2 \frac{N}{\log N} \}$ $N \to \infty$ principal exponential order. When $d\geq 4$, these tails...

We consider percolation of the vacant set random interlacements at intensity $u$ in dimensions three and higher, derive lower bounds on truncated two-point function for all values $u>0$. These are sharp up to principal exponential order dimension $u \neq u_\ast$ higher dimensions, where $u_*$ refers critical parameter model, they match upper derived article arXiv:2503.14497. In three, our results further imply that grows large distances $x$ a rate depends only through its Euclidean norm,...

We consider the Gaussian free field on $Z^d$, $d\geq3$, and prove that critical density for percolation of its level sets behaves like $1/d^{1+o(1)}$ as $d$ tends to infinity. Our proof gives principal asymptotic behavior corresponding $h_*(d)$. Moreover, it shows a related parameter $h_{**}(d)$ introduced by Rodriguez Sznitman in [23] is fact asymptotically equivalent

The Discrete Gaussian model is the lattice free field conditioned to be integervalued.In two dimensions, at sufficiently high temperature, we show that scaling limit of infinite-volume gradient Gibbs state with zero mean a multiple field.This article second in series on model, extending methods first paper by analysis general external fields (rather than macroscopic test functions torus).As byproduct, also obtain for mesoscopic torus.

We study the asymptotic behaviour of random integer partitions under a new probability law that we introduce, Plancherel-Hurwitz measure.This distribution, which has natural definition in terms Young tableaux, is deformation classical Plancherel measure appears naturally context Hurwitz numbers, enumerating certain transposition factorisations symmetric groups.We regime number factors underlying grows linearly with order group, and corresponding topological objects, maps, are high genus.We...

Abstract We consider a percolation model, the vacant set of random interlacements on , in regime parameters which it is strongly percolative. By definition, such values pinpoint robust subset supercritical phase, with strong quantitative controls large local clusters. In present work, we give new characterization this terms single property, monotone involving disconnection estimate for . A key aspect to exhibit gluing property clusters from information alone, and major challenge undertaking...

For lattice systems of statistical mechanics satisfying a Lee-Yang property (i.e., for which the circle theorem holds), we present simple proof analyticity (connected) correlations as functions an external magnetic field h, Re h > 0 or < 0. A survey models known to have is given. We conclude by describing various applications aforementioned in h.

Abstract We investigate level sets of the Gaussian free field on continuous transient metric graphs $$\widetilde{{\mathcal {G}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>G</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:math> and study capacity its set clusters. prove, without any further assumption base graph $${\mathcal {G}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> , that sign clusters is finite almost surely. This...

For the Bargmann–Fock field on Rd with d≥3, we prove that critical level ℓc(d) of percolation model formed by excursion sets {f≥ℓ} is strictly positive. This implies for every ℓ sufficiently close to 0 (in particular nodal hypersurfaces corresponding case ℓ=0), {f=ℓ} contains an unbounded connected component visits "most" ambient space. Our findings actually hold a more general class positively correlated smooth Gaussian fields rapid decay correlations. The results this paper show behavior...

We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines canonical percolation model with strong, algebraically decaying correlations. prove that three natural critical parameters associated to model, respectively describing well-ordered subcritical phase, emergence an infinite cluster, and onset local uniqueness regime in supercritical actually coincide. At core our proof lies new...

For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters Gaussian free field on $G$ fall into regime strong supercriticality, in which two infinite dominate (one for each sign), and finite are necessarily tiny, with overwhelming probability. Examples belonging to this include regular lattices like $\mathbb{Z}^d$, $d \geqslant 3$, but also more intricate geometries, such as Cayley suitably growing (finitely generated) non-Abelian groups, cases random walks...

The Discrete Gaussian model is the lattice free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on torus a multiple of field. Our proof starts from single renormalisation group step after which integer-valued becomes smooth then analyse using method. This paper also provides foundation for construction infinite-volume gradient Gibbs state in companion paper. Moreover, develop all estimates general...

The Discrete Gaussian model is the lattice free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that scaling limit of infinite-volume gradient Gibbs state with zero mean a multiple field. This article second in series on model, extending methods first paper by analysis general external fields (rather than macroscopic test functions torus). As byproduct, also obtain for mesoscopic torus.