111 problems

Consider a square lattice at the critical percolation point $p_c=1/2$, and its medial lattice with wired boundary conditions on one side and free boundary conditions on the other.…

Let $b\in\mathbb{R}$, and define $a=a(b)>0$ by … Set $\kappa=2/a^2$ and let $\kappa'=\kappa(-b)$, so that $\kappa\kappa'=16$. Duplantier's SLE duality conjecture. When $\kappa>4$,…

Let $\mathcal{J}^{(m,n)}_{\alpha}$ be the Coulomb gas integral associated with a $(n,m)$ chordal link pattern $\alpha$, where $m\leq (n+2)/2$. These functions satisfy the null vect…

Let $D\subset\mathbb{C}$ be a simply connected domain, let $x,y$ be distinct boundary points, and let $\eta$ be a chordal curve from $x$ to $y$. A random curve from $x$ to $y$ is r…

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

A self-avoiding walk (SAW) is a contiguous sequence of moves on a lattice that does not cross itself and does not visit the same point more than once. In two dimensions, consider t…

Let $\mathcal{K}^{(m,n)}_{\alpha}$ denote the screening solution constructed by Coulomb gas integrals from a chordal link pattern $\alpha$, in the presence of a marked boundary poi…

Let $\mathcal{J}(\boldsymbol{x},u)$ be a screening solution obtained by integrating the Coulomb gas integrand over an admissible choice of screening contours. These functions satis…

Let $\mathcal{Z}_{\alpha}(\boldsymbol{z},u)$ and $\mathcal{Z}_{\alpha}(\boldsymbol{x})$ be pure partition functions, and let $\Phi(\boldsymbol{x},\boldsymbol{\xi},u)$ and…

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 ${\bf a}$…

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 $\gamma(t)$ 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…

Self-avoiding walks are lattice paths that visit no site more than once. The Schramm–Loewner evolution with parameter $\kappa=8/3$, denoted $\operatorname{SLE}_{8/3}$, is a random…

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 $\operatorname{SLE}_{8/3}$ denote Schramm–Loewner evolution with parameter $\kappa=8/3$, and let self-avoiding walks be lattice paths that do not visit the same vertex more tha…

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 $(K_{\tau_u},W_{\tau_u})_u$ be the stable Lévy process associated with the cut-time parametrization in the source. For $\kappa\geq 6$, its restriction formula is … where $A$ is…

Let $\operatorname{SLE}(\kappa,\kappa-4)$ be the process described above, with $<\kappa$ and let $\kappa'$ satisfy … Run the process until infinity and consider its right boundary,…

Dubédat's boundary duality conjecture. The right boundary of $SLE^\sigma(\kappa,\kappa-4)$ is an $SLE(\kappa',(\kappa'-4)/2)$. The conjecture is a precise proposed realization of S…

Let $\mathbb H$ be the upper half-plane and let $\gamma$ be a chordal $SLE_\kappa$ path from the origin to infinity, with fixed $\kappa\leq 8$. SLE reversibility conjecture. The la…

Let $\gamma$ be a very long rescaled self-avoiding walk in the half-plane, let $H$ be a suitable subdomain, and let $\Phi_H$ be the conformal map used in the half-plane restriction…

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 $\phi$ be a height function on $\mathbb Z^2$, let $\overline{\phi}$ be its continuous piecewise-linear interpolation on simplices, and define the level sets…

Let $\kappa\leq 8$, and let the $SLE_\kappa$ generating curve be the random curve generated by Schramm–Loewner evolution with parameter $\kappa$. Its Hausdorff dimension is known t…

Let the Brownian frontier denote the outer boundary of a planar Brownian curve, and let the scaling limit of long self-avoiding curves be the conjectural continuum limit of self-av…