196 problems

For a planar lattice $\Lambda$ in Euclidean space, let $K_\Lambda$ denote the lattice-dependent constant governing the asymptotic critical-temperature scaling under iterative trian…

Recurrence conjecture. The simple random walk on the $\nu$-IIPT is almost surely recurrent for every value of $\nu$.

Let $d\geq 5$ and consider a discrete torus of side length $L$. For the Ising model and self-avoiding walk, let their two-point functions be defined in the usual way, and let…

Let the Ising model have short-range interactions on the lattice in spatial dimension $d$, with spins taking values in $\\{-1,1\\}$. A higher-dimensional absence-of-transition conj…

Consider the high-temperature Ising model and critical Bernoulli percolation as two planar lattice models. High-temperature Ising–percolation universality conjecture. The universal…

Consider the planar Ising model with Dobrushin boundary condition in the canonical ensemble with $m=0$, and let the magnetic field be the field given by $h_N$ in the source. Rescal…

Let $d>4$, let $\Lambda_n$ be the finite torus used for the critical Ising model with periodic boundary conditions, and set $\beta_c=\beta_c(d)$. Let $\bar{Y}_n$ be the normalized…

Wilson's conjecture. The scaling limits of these XOR-Ising loops can be described as the union of level sets of the Gaussian Free Field with an appropriately tuned spacing.

Lee–Yang's conjecture. The magnetization satisfies

Right-divisibility conjecture. This property holds for all values of $n$. The conjecture proposes that the observed nested factorization of the differential operators annihilating…

Consider the Ising model on a tree at zero external field, with Glauber dynamics started from the symmetric product measure $\nu={{\mathbb P}}_{1/2}$. Write $\nu P_t$ for its law a…

Consider the Ising model and let ${\mathcal B}\subset [\beta_c,\infty[$ be the at most countable set of inverse temperatures at which the pressure may fail to be differentiable. Is…

Consider the Ising model on the regular tree $T_b$. A local reconstruction algorithm is one in which each vertex scans the information stored at its descendants $\ell$ generations…

High-temperature spectral-branch conjecture. Results analogous to those proved for the one-dimensional model should hold for sufficiently small $\beta$ in dimensions $d\geq 2$.

Subexponential relaxation-time conjecture. A weaker conjecture is

Critical logarithmic relaxation-time conjecture. The relaxation time is of order

The model is the ferromagnetic Ising Glauber dynamics on a finite tree or lattice region with $n$ vertices, at zero external field $h=0$ and inverse temperature $\beta>\beta_c$, wh…

Let $\beta>\beta_c$, $h\geq h_w(\beta)$, and $\rho>0$. For each $N$, let $\mu_{\beta,\lambda,h,\Lambda_N}$ be the Ising-model measure in the box $\Lambda_N$ with external-field par…

Let $\Omega$ be a bounded open set in $\mathbb{R}^3$ with smooth boundary, and let $\gamma$ be a Jordan curve on $\partial\Omega$ separating it into relatively open sets $\Gamma^+$…

Consider the two-dimensional Ising model with boundary condition … denote the set of infinite-volume translationally invariant Gibbs measures, and consider the finite-volume Gibbs…

Finite-temperature limit-point conjecture. Any convex combination of $\mu^+$ and $\mu^-$ is a limit point of

Let $\hat{S_0}$ denote the ground state entropy density of the Ising model, and consider square lattices with toroidal boundary conditions. Loebl and Vondrák obtained an upper boun…

Faceted-droplet hypothesis. As $l\to\infty$, with probability tending to $1$, there exist six distinct two-dimensional planes $L_i=L_i(\sigma)\subset\mathbb{T}^3$, $i=1,\ldots,6$,…

Let $s=\sinh 2\beta J$ be the complex temperature variable, and let the susceptibility be regarded as a function of $s$. Its finite-lattice singularities lie on the unit circle…

Threshold conjecture. The critical $X=X_0$ is effectively a threshold beyond which the signal becomes effective.