55 problems

Conformal invariance conjecture. The scaling limit of the model is conformally invariant.

Let $D$ be a simply connected domain in the complex plane, and let $a,b,c,x$ be boundary points in counterclockwise order. Suppose that $D$ is conformally equivalent to the equilat…

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

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

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

Aizenman's conformal invariance conjecture. The scaling limits of critical percolation interfaces are conformally invariant and their crossing probabilities satisfy Cardy's formula…

Let $b$ and $C$ be as in the conjectured asymptotic … After multiplying the union over boundary endpoints by $N^{2b-2}C^{-1}$ and scaling paths by $N^{-1}$, consider excursions in…

Let $D$ be a domain with marked boundary points $A$ and $B$, and let a random curve $\gamma$ join $A$ to $B$. If the initial segment $\gamma[0,t]$ is known, let $D_t$ be the connec…

Let $D$ be a fixed open subset of the complex plane, and let $D_\delta$ be a discrete approximation at mesh size $\delta$ of $D$. Let the corresponding critical loop or cluster mea…

Let $\kappa>4$ and set $\kappa'=16/\kappa$. In the configuration $(\mathbb{H},x,y,z,\infty)$ with $x<y<z$, let $(K_t)$ be the Loewner chain obtained by running an…

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…

Let the c3-finite bead process be conditioned to survive up to distance $r$, equivalently up to time $r^2$ or imaginary part $r$, and let $r\rightarrow\infty$. Scaling limit conjec…

Self-avoiding-walk SLE conjecture. If the scaling limit of self-avoiding walk exists and is conformally invariant, then its only possible limit is $SLE_{8/3}$; the source further r…

Self-avoiding-walk dimension conjecture. It is conjectured that the scaling limit of planar self-avoiding walks has paths of dimension $4/3$.

Let critical percolation on a lattice have a scaling limit in a planar domain $D$, with connections interpreted as the continuum connectivity events. For two domains $D$ and $D'$ r…

Consider critical bond percolation with $p=p_c=1/2$ in the discrete half-plane $\mathbb{Z}\times\mathbb{N}$, with the specified boundary edges erased for $x\geq 0$ and open for…

The scaling limits of many two-dimensional systems in statistical physics are expected to exhibit conformally invariant behaviour at criticality. Conformal invariance conjecture. A…

Uniqueness conjecture. For every $0<\theta\leq1$, the LCC assignment whose existence is asserted in Theorem 1 is unique up to multiplication by a scalar factor.

In the two-dimensional continuum limit, let ${\cal R}$ be a finite simply connected region of the plane bounded by a curve $\Gamma$, and let ${\cal L}_a$ be a sequence of lattices…

In Yang–Mills scattering amplitudes, consider the sign factors arising when amplitudes are Fourier transformed from momentum space to real twistor space for split-signature space-t…

Conformal-coordinate conjecture. For more general boundary conditions and other periodic lattices, an appropriate analogue of the critical point mapping gives the correct coordinat…

For each $\kappa' \in (4,8)$, consider critical FK percolation with cluster weight … Its loop ensemble consists of the outer and inner boundaries of FK clusters, alternating betwee…

Arc-loop-ensemble scaling-limit conjecture. As the mesh of the planar domain tends to zero, the full collection of double-dimer arcs should converge in distribution to the arc loop…

Consider the loop $O(n)$ model with $n\in (0,2]$ and boundary condition $\beta\in\mathrm{LP}_N$. Let $\varsigma^{\delta}$ be the random connectivity in $\mathrm{LP}_N$ induced by t…

Nienhuis's conformal-invariance conjecture. The model exhibits conformal invariance for every $x\geq x_c(n)$. This conjecture concerns the expected conformally invariant behaviour…