46 problems

Consider the $|\varphi|^4$ model with free boundary conditions (FBC), and let $\chi_{g,\nu,N}^{\rm F}$ denote its susceptibility, while $\chi_{g,\nu,N}$ denotes the susceptibility…

Let $d>8$, let $\mathbb{T}_r^d$ be the $d$-dimensional torus of volume $V$, and let $I_0(s)=\int_0^\infty e^{-t^4/4+st^2/2}\,dt$. Let $G_p^\mathbb{T}(x)$ be the torus two-point fun…

Consider critical percolation in a box of width $r$ in dimension $d>6$, and let $V=r^d$. Under free boundary conditions, clusters are connected only within the box; under periodic…

In subcritical percolation, let $G_N(z)$ denote the distribution function of the rescaled largest-cluster size, and let its envelope functions be … Assume these envelope functions…

Let $E_L$ and $C_L$ denote the finite-size correction quantities introduced in the source for strips of width $L$, for both the square and triangular lattices. Vanishing-correction…

Let $q_0(L)$ be the crossing value for an even-width square-lattice strip, with finite-size behavior fitted by $q_0(L)=q_{\rm c}(\mathrm{sq})+A L^{-1/\nu}$. Square-lattice exponent…

Let $\operatorname{\bf NG}_{s}$ be the probability measure whose moment generating function is specified by … Under the assumptions of Theorem, let $w_N$ denote the critical-window…

For the $(\varepsilon,p)=(1,1)$ chain, consider the continuum tower in the spin sector $[\mathcal{S}^{(\ell)},\mathcal{S}^{(r)}]=[0,2]$. Let $\mathit{n}$ be the tower excitation nu…

For the $D^{(2)}_3$ chain with boundary conditions $(\varepsilon,p)=(1,1)$, let $\gamma$ satisfy … In the Cartan sector $(h^{(\ell)},h^{(r)})$, let $N$ be the lattice size and let…

For the $D^{(2)}_3$ chain with boundary conditions $(\varepsilon,p)=(1,0)$, let $X_{\rm eff}(h_1,0)$ be the limiting effective scaling dimension in the sector with Cartan charges…

For the $D^{(2)}_3$ open chain with boundary conditions $(\varepsilon,p)=(0,0)$, let $h_1$ and $h_2$ denote the Cartan charges, let $\gamma$ be the anisotropy, and let…

Pseudocritical-profile conjecture. For $d>d_{\mathrm{c},\alpha}$, there is a pseudocritical point

Table-profile conjecture. The profile functions $f(s)$ are the functions listed in Table for both SR and LR models in every dimension $d\ge d_{\mathrm{c},\alpha}$, including the up…

Universal-profile conjecture. The preceding susceptibility and two-point-function asymptotics hold for every listed model, with the same profile function for its SR and LR versions…

Universal-profile conjecture. The profiles computed for self-avoiding walk and branched polymers on the complete graph should also apply to the corresponding SR and LR models at an…

Percolation scaling-profile conjecture. There exist constants $a,b>0$ such that, as $V\to\infty$,

Lattice-tree and lattice-animal profile conjecture. The same critical finite-size scaling profile $I_0$ occurs for both lattice trees and lattice animals on $\mathbb{T}_r^d$ above…

Consider Bernoulli bond or site percolation in dimension $d>6$ on the box $\Lambda_r$, using only bonds or sites in the box for free boundary conditions. Let $p_c$ be the infinite-…

Let $W^*$ be the Brownian excursion of length $1$, let $\Psi(x)=\mathbb{E}\exp\left[x\int_0^1W^*(t)\,\mathrm dt\right]$ be the moment-generating function of its area, and define ……

Free-boundary effective critical point conjecture. There exists such a $\beta_{c,r}$ for which all statements of the non-Gaussian limit and universal profile conjecture hold with…

Non-gaussian limit and universal profile conjecture. As $r\to\infty$, $h_r^{-1}S_r$ converges in distribution under $\langle\cdot\rangle_{\beta_c+sw_r}^{\mathbb{T}_r}$ to…

The twisted amplitude conjecture. The amplitude is

The amplitude conjecture. The amplitude is

At the critical dimension $d=3\alpha$, compare hierarchical percolation on $cLambda_n$ with free and periodic boundary conditions, where $L$ is the hierarchical scale parameter. Pe…

Let $Q_0(\tau,N)$ denote the persistence probability at time $\tau$ for matrices of finite size $N$, and let $\mu$ be the persistence exponent from the large-$N$ asymptotic…