27 problems

Let $D$ be an open bounded simply connected domain of $\mathbb{C}$. Let $D^\varepsilon$ be a fine-mesh approximation of $D$, $A^{\varepsilon}$ the approximation of the boundary…

Corrector Gaussian free field conjecture. Let $d\ge1$. Then the law of

Ashkin–Teller magnetisation-field decomposition conjecture. The statement of the main magnetisation-field decomposition theorem holds for the Ashkin–Teller magnetisation field at e…

Let $\mathsf{Ht}_\mathcal N$ be the random height function of the model at scale $\mathcal N$, let $\mathbb E[\mathsf{Ht}_\mathcal N]$ be its expectation, let $\pounds$ denote the…

Gaussian free field–discrete Gaussian conjecture. The fluctuations of the dimer-model height function in the liquid region of a graph of scale $N$ are approximated in distribution…

Let $D$ be a fixed bounded domain, let $h$ be the Gaussian free field on $D$, and let $(\boldsymbol{\lambda}_n)_{n\ge 0}$ be the eigenvalues of Liouville Brownian motion associated…

Let $\hat{\mathcal G}$ be obtained by adjoining to a graph $\mathcal G$ its reflection across the real axis while retaining one copy of vertices on the real axis. Let $M$ be an inf…

For any $z>0$, superimpose two independent free boundary dimer models in the upper half-plane, and consider the resulting collection of boundary-touching curves. Free-boundary dime…

Near-critical scaling-limit conjecture. As $a\downarrow0$,

Let $T_{\mathrm{rec}}^+$ denote the upper reconstruction threshold for the statistical reconstruction problem considered in the paper. Upper-threshold conjecture. … The conjecture…

Let $\phi_n$ be a zero-boundary discrete Gaussian free field converging to a continuum Gaussian free field $\Phi$, and let $\mathbb{A}_{-\lambda,\lambda}$ denote the continuum ALE.…

Let $\phi_n$ be the discrete Gaussian free field, let $u_n$ be the auxiliary function appearing in the construction, and let $\eta^{(n)}$ be the original level line. For sufficient…

Let $T_{\mathrm{rec}}^-$ denote the lower reconstruction threshold for the statistical reconstruction problem considered in the paper. Lower-threshold conjecture. … The claim is pr…

Compactified Gaussian free field conjecture. The limiting height form is given by the compactified Gaussian Free Field in the appropriately punctured surface.

Fix an arbitrary tilable polygonal domain $\Upsilon$ on the triangular grid. Let $H_L(x,y)$ be the random height function of a uniformly random lozenge tiling of the dilated domain…

Conjectural two-dimensional fluctuation theorem. The centered random height function, multiplied by $\sqrt{\pi}$, converges in distribution to the pullback under $\Omega$ of the Ga…

First-passage extremal-distance conjecture. The extremal distance $\operatorname{ED}(\Lambda_a^\Sigma,B_2(\Sigma))$ is distributed as the last visit time of level $a$ by a Brownian…

Let $X$ be the normalized exponential functional introduced in the source, and let $q$ be in the range where the displayed moments exist. Freezing hypothesis. For $\beta>1$, … This…

Let $V_N$ be the maximum of the discretized Gaussian free field on the circle with singularity parameter $\alpha\geq 0$. Let $M_{(\tau=1,\alpha,\alpha)}$ denote the critical Morris…

Quantum-measure determination conjecture. In the setting described above, the measure $cmu_{X,h}$ almost surely determines the harmonic extension of $h$ off $X$.

Let $(X_n)_n$ be a cutoff approximation of a centered Gaussian field on $[0,1]^d$ with logarithmic covariance, and set $c_n={\rm Var}(X_n)$. Let $M'$ be the derivative chaos and le…

Wilson's conjecture. The scaling limit of the family of loops of the critical XOR Ising model are the level lines of the Gaussian free field corresponding to levels that are odd mu…

Let $\mathbf{m}_n\in\mathbf{M}_n^p$ be sampled according to a statistical physics model in the $\gamma$-universality class. Let $\widehat{\mathbf{m}}_n\subset\mathbb{S}^2$ be any n…

The parameters $\omega_n$ define models of random partitions, and $\omega_n$ converges to $0$ at speed $1/\sqrt{n}$ as $n$ tends to infinity. The fluctuations of the partitions are d…

Tightness conjecture. The sequence of random variables $\{Y_N\}_{N\geq 1}$ is tight.