101 problems

Let $\theta_n$ satisfy the scaling condition in the source, and let $X_n=(X_n(k);k=0,1,\dotsm)$ be the breadth-first walk on the random graph $\mathscr{G}_n^\theta$ constructed for…

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…

Aldous's conjecture. As $q\downarrow 0$, the rescaled stationary East process converges to a limit point process $X_t$ on $[0,+\infty)$ such that:

Self-avoiding loop scaling-limit conjecture.

Consider a graph obtained by a non-trivial stitching of the lattices $\mathbb{Z}^{d}$ and $2\mathbb{Z}^{d}$. Stitching isotropy conjecture. Any such graph is an isotropic graph. If…

Quasi-loop theorem conjecture. Theorem 4.6 should hold for any Euclidean net. Extending the result would give quasi-loop control for loop-erased random walks on all Euclidean nets,…

Long-range percolation scaling conjecture. If $\alpha\in(0,2)$, then the corresponding random walk scales to a symmetric $\alpha$-stable Lévy process in $\mathbb R^d$.

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

Uniform spanning tree scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that (rhoscal) holds for $\rho^{(\mathrm{st})}$.

Invasion percolation scaling conjecture. For each $r\geq2$, $\vec t\in\mathbb{R}^{r-1}$ and $\vec k\in\mathbb{R}^{d(r-1)}$, there exist constants $A,V,v$ and $\delta\in(0,1)$ such…

Incipient infinite lattice tree scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that (rhoscal) holds for $\rho^{(\mathrm{lt})}$.

Unoriented percolation scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that the scaling relation (rhoscal) holds for $\rho^{(\mathrm{pe})}$.

Let $\gamma$ denote a rescaled self-avoiding curve, and let $SLE_{8/3}$ denote chordal Schramm–Loewner Evolution with parameter $8/3$. Self-avoiding-walk scaling-limit conjecture.…

Let $D$ be a domain with boundary points $A$ and $B$, and let $P_{D,A,B}^{SAW}$ denote a scaling limit of measures on self-avoiding curves from $A$ to $B$ in $D$, assigning weight…

Consider the random K-LSAT linear program with $n$ variables and $m$ constraints, with all auxiliary variables $\psi$ set to zero. Let $N(n,m)$ be the maximum cardinality of a feas…

Let $N(n,m)$ be the maximum number of clauses that can be satisfied by one assignment of $n$ variables in a random K-SAT instance with $m$ clauses, and let $c^*_K$ be the threshold…

Let ${\bf X}^0(t)$ be the East process on the positive integers, started from a single particle at site $0$. Let $r({\bf x})$ be the rightmost occupied site in configuration…

Let $G_n$ be the partially wired square grid described in the source, let $\omega$ have the critical random-cluster measure with $q\in(0,4)$, let $\omega^\dagger$ be its dual confi…

Let $\mathbb{U}$ be the unit disk and consider a square grid of mesh $\epsilon$. Let the uniform measure be taken on simple grid paths from $0$ to $\partial\mathbb{U}$, and separat…

CFT scaling-limit conjecture. The scaling limits of two-dimensional statistical mechanics models undergoing a continuous phase transition are given by the correlation functions of…

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…

Let $\hat\tau^{(n)}$ denote the first detachment time in the $n$-detachment process. Scaling-limit conjecture. The random variables … have a limiting distribution as $n\to\infty$.…

Let $R_n$ be the resistance in the critical resistance model, let $p_c=\frac12$, and let $c_{\mathrm{eff}}$ be the constant in the critical scaling conjecture for $\log R_n$. Derri…

Let $R_n$ denote the resistance in the random resistance model, let $p_c=\frac12$, and let $y\in\mathbb R$. Addario-Berry et al.'s conjecture. If $p=p_c=\frac12$, there exists a co…

For $\gamma\in(0,2)$ and $\widetilde\theta\in(0,\pi)$, let $\mathfrak L_\Phi^{\operatorname{LDP}}$ and $\mathfrak L_\Phi^{\operatorname{SDP}}$ denote candidate random functions obt…