404 problems

The level lines of the ${\mathbb Z\textsc{gff}}$ measure near the corner of the limit shape have fluctuations on the scale $O(L^{1/2})$. Brownian scaling conjecture. Their scaling…

Let CLE$_4$ denote the conformal loop ensemble with parameter $4$. The double-dimer model is a random collection of loops, and the loop $O(n)$ model is parametrized here by $n=2$ a…

In two dimensions, let self-avoiding walk be rescaled in the usual way, and let $\operatorname{SLE}_{8/3}$ denote Schramm–Loewner evolution with parameter $8/3$. Lawler–Schramm–Wer…

Schramm's conjecture. LERW has a conformally invariant scaling limit characterized by Schramm-Loewner evolution with parameter $2$.

Diffusive-scaling conjecture. The laws of the processes $\mathcal{M}^{(m)}$, as $m\to\infty$, have a limit. The process has monotone non-decreasing paths, and the conjecture asks f…

Super-Brownian scaling conjecture. The critical-dimensional model should have the same super-Brownian scaling limit as the high-dimensional model once appropriate logarithmic corre…

Liouville quantum gravity scaling-limit conjecture. The limit in law as $a\to0$ of $( ua,nua^{partial})$ exists in the space of Radon measures equipped with the topology of weak co…

Self-avoiding loop scaling-limit conjecture.

Let $\kappa\in(4,8)$, and consider a statistical mechanics model in two dimensions whose cluster converges to the ${\rm CLE}_{\kappa}$ gasket. Equip the discrete cluster with its e…

Consider the Gaussian free field $\varphi$ on $\mathbb{Z}^d$, with $d\in\{3,4,5\}$. Let $\{\varphi\geq0\}$ be its nonnegative excursion set, and consider the clusters obtained from…

Potts-to-simple-CLE conjecture. The scaling limit of the critical $q$-state Potts model can be described by simple $\mathrm{CLE}_\kappa$. The conjecture identifies simple CLE as th…

Consider the planar Ising model with Dobrushin boundary condition in the canonical ensemble with $m=0$, and let the magnetic field be the field given by $h_N$ in the source. Rescal…

Cardy's formula. The scaling limit exists and equals the conformal coordinate of $X$:

Let $d>4$, let $\Lambda_n$ be the finite torus used for the critical Ising model with periodic boundary conditions, and set $\beta_c=\beta_c(d)$. Let $\bar{Y}_n$ be the normalized…

Let $f_j^N$ be the $j$-particle marginal, with initial data $f_j^N(0)=f_0^{\otimes j}$, where $f_0$ is a given one-particle density. The difficulty in deriving the Landau equation…

Let $h$ be a variant of the Gaussian free field on a planar domain, let $dx^2+dy^2$ be the Euclidean metric tensor, and let $\gamma\in(0,2)$. The associated Liouville quantum gravi…

Wilson's conjecture. The scaling limits of these XOR-Ising loops can be described as the union of level sets of the Gaussian Free Field with an appropriately tuned spacing.

Let $d\geq 3$ and let $\phi^{(\xi)}$ be the corrector. For $\varepsilon>0$, define the random distribution … for $f\in C_c^\infty(\mathbb{R}^d)$. Gaussian free field scaling-limit…

Scaling-limit conjecture. As $\delta\to0$, the measures ${\bf P}_{D,\delta}$ converge to a probability measure ${\bf P}_D$ on simple paths from the origin to $\partial D$, with

Let $D\subset\mathbb{C}$ be a bounded domain with smooth boundary, let $z,w\in\partial D$ be distinct, and let the lattice spacing be $\epsilon$. At the critical parameter…

Let $N_\varepsilon(\mathcal F)$ denote the number of points of branching of scale $\varepsilon$ in a scaling-limit spanning-tree configuration $\mathcal F$. The branching number of…

Let $\gamma(t)$ be a random curve arising as the scaling limit of a critical two-dimensional lattice model, with the lattice length suitably scaled to give its natural parametrizat…

Marckert–Mokkadem conjecture. The scaling limit of random quadrangulations should be given by the Brownian map.

Let $G_n$ be a sequence of finite graphs satisfying either the conditions of Case 1 stated above or $G_n=\mathbb{Z}_n^4$. Consider the length of the loop-erased random walk on…

Consider the FK cluster model on a planar lattice, with cluster weight $q$ and edge parameter $\beta$, and let $\beta_c$ be its critical value. The edge subsets determine collectio…