27 problems

Loop $O(n)$ exponential-decay conjecture. The loop $O(n)$ model exhibits exponential decay of correlations when $n>2$.

Potts critical-temperature conjecture. For the Potts model, the critical inverse temperature behaves as

Computation-tree correlation-decay conjecture. Correlation decay on the computation tree holds in this regime, at least when the number of colors $q$ is constant.

Low-alpha correlation-decay conjecture. Correlation decay on the computation tree holds for much lower values of $\alpha$.

Potts correlation-decay conjecture. Correlation decay can in fact be established in the regime

Let $\nu=\nu(\beta,q,d)$ be Potts lattice gauge theory on $\mathbb{Z}^d$. Let $\sigma_N=[0,1]^2\times\{N\}\times\{0\}$ be a plaquette, and for a plaquette $\sigma$ let…

For the Sine$_\beta$ point process at $\beta=2$, the two-point truncated correlation function is … A Berezinskii–Kosterlitz–Thouless transition conjecture. The critical value…

Iagolnitzer–Souillard conjecture. This asymptotic behavior should hold whenever $h\neq 0$.

Let $\mathcal{K}^{\mathrm{barcode}}_{(M)}(0,t)$ denote the finite-truncation barcode kernel, and let $t\in\mathbb{Z}_{>0}$. Exponential correlation-decay conjecture. The correlatio…

Consider the high-dimensional integration setting in which the presented method approximates a Riemann sum of the target function within a factor of $1+O(1/n)$ in time…

A polytope is a convex polytope, and the correlation decay property is the property that correlations between assignments to vertices of a graph decay exponentially with their grap…

Let $P$ be a polytope, and let the Gibbs distribution be the probability distribution associated with the Lebesgue measure on $P$. Correlation decay conjecture. The Gibbs distribut…

Let $h$ denote the external magnetic field in the Spin $O(N)$ model, where $N$ is an integer, and let the decay constant be the constant in the exponential rate of decay of spin-sp…

Let $J_x$ be the coupling constants, let $u{beta}(s)$ denote the inverse correlation length in direction $s$, and let $beta{rm c}$ be the critical inverse temperature. **Exponenti…

For a sequence of decorated graphs $G_n=(V_n,E_n,\mathcal{A}_n)$ and distances $R_n\to\infty$, asymptotic SSM at distances $R_n$ means … The paper's main theorem proves its claimed…

Let $\mathcal{F}$ be the family of all $d$-regular trees, and consider the $K$-coloring model. Spatial Mixing (SM) concerns decay of the influence of boundary conditions on local m…

The paper studies partition functions represented by graph polynomials and the relation between the absence-of-complex-zeros property and correlation decay. For amenable graph fami…

Consider planar Sinai billiard flows whose boundary may have corner points, with the angle between the intersecting boundary pieces at each corner nonzero. Corner-point correlation…

Let $K_n$ be the upper bound on the number of smooth pieces into which the $n$th iterate of a smooth convex expanding submanifold is chopped by the billiard map. Complexity hypothe…

Let $\beta<8\pi$, and consider the formally massless continuum sine-Gordon model in the limit $L\to\infty$ and $m^2\to 0$ with nonzero coupling parameter $z\neq 0$. Its equilibrium…

Let $\Lambda\uparrow\mathbb{Z}^d$ with $d\geq 3$, and consider the $\mathbb{H}^{2|2}$ model with external field $h>0$. Suppose the spin-spin correlations decay exponentially at a r…

Let IQHE and FQHE denote the integer and fractional quantum Hall effects, respectively. Suppose that correlations of all local observables decay faster than any exponential. Correl…

Let $SO(N)$ be the group of $N\times N$ orthogonal matrices with determinant $1$, let $\Lambda\subset\mathbb{Z}^d$ be finite, and let $\mu_{\Lambda,N,\beta}$ be the lattice gauge-t…

Exponential-decay conjecture. This correlation function has exponential decay with respect to $\|x-y\|$.

The first function considered is the energy of a packet of low-frequency modes in the FPU problem, and its time autocorrelation is the quantity under consideration. Sub-exponential…