38 problems

Friedgut–Kahn–Kalai–Keller's off-diagonal conjecture. One should have

Simon-type conjecture. The displayed inequality holds for every $q\in[1,2]$.

Let $K_1,K_2\subseteq\mathbf{R}^n$ be convex sets whose centroids with respect to the standard Gaussian measure $\mu_n$ coincide. Non-symmetric Gaussian correlation conjecture. Doe…

The directed correlation conjecture. On $Q$, the inequality

Let $c(F)$ be a separable Banach space and let $cmu$ be a centered Gaussian measure on $F$. Gaussian correlation conjecture. [The supplied candidate is incomplete and contains no a…

Kalai–Keller–Mossel correlation conjecture. For every increasing and balanced family $\mathcal{A}\subseteq\Omega_n$, there exists an increasing linear-threshold family…

Logarithmic-shift BK conjecture. The KPZ line ensemble satisfies a form of the BK inequality with a shift that is logarithmic in the size of the interval over which the first curve…

Let $d<6$ and consider the critical FK-Ising model on $\mathbb Z^d$. The four-point function refers to the four-point connectivity quantity controlled by the correlation inequaliti…

Let $G$ be a graph and let $a,b,c,d$ be vertices of $G$. For the arboreal gas measure $\mu^{\text{F}}_{G,\lambda}$, the notation $\mu^{\text{F}}_{G,\lambda}(a,b)$ and similar expre…

Let $G=(V,E)$ be a graph and let $u,v,w\in V$. Let $\mu^{\text{F}}_{G,\lambda}$ denote the arboreal gas measure with parameter $\lambda$, and write…

Let $\mu$ be an ERGM measure on graphs with vertex set $[n]$, and let $e_1,e_2,e_3$ be edges spanning a triangle in $K_n$. Let $e$ be an arbitrary edge. Triangle moment conjecture.…

Kahn's conjecture. If $|X|=|Y|=|Z|$, then

Extremal intersection conjecture.

Monotonicity conjecture. For $\beta_1,\beta_2\in(0,1)$ with $\beta_1<\beta_2$, there exists $\varepsilon>0$ such that

Let $P$ be a finite poset with $n$ elements, let $x\in\min(P)$ and $y\in\max(P)$, and let $e(P)$ denote the number of linear extensions of $P$. Endpoint deletion lower-bound conjec…

Let $P=(X,\prec)$ be a finite poset, and fix distinct elements $x,y,z\in X$. For $k,\ell\geq1$, define … and let $\operatorname{F}(k,\ell)=|\mathcal F(k,\ell)|$. Cross-product conj…

Let $n\in\mathbb{N}$ and let $p_1,\dots,p_n\in 2\mathbb{N}$. For a real centered Gaussian vector $(X_1,\dots,X_n)$, the statement $\mathbf{GPI}_n(p_1,\dots,p_n)$ means … with equal…

CGKS 2 conjecture. For all integers $N\ge 3$, the $O(N)$ model satisfies the CGKS 2 inequalities, and for all integers $N\ge 2$, the $\mathbb{C}\mathbb{P}^{N-1}$ model satisfies th…

Sahi's coefficient-positivity conjecture. For every $n\geq 1$, the coefficient $c_n$ satisfies

Let $L$ be a distributive lattice equipped with a probability measure $mu$ satisfying … for all $a,b\in L$; such an $L$ is an FKG poset. Let $f_1,\ldots,f_n$ be positive monotone d…

One-half condition for the GHS inequality. In the general setting, the GHS inequality holds if and only if

Let $W$ be a Gaussian process, let $I$ be a countable index set, let $(s_j,t_j)_{j\in I}$ be a collection of intervals, and let $(\varepsilon_j)_{j\in I}$ be positive bounds. For d…

Let $M_n$ be the finite honeycomb-lattice region. For $k=1,2,3$, choose sets $A_k,B_k\subset M_n$, and let $U_k$ be the event that fluid $k$ percolates between $A_k$ and $B_k$. Gen…

Let $M_n$ be the finite honeycomb-lattice region, let $O$ be its central cell, and let $A_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates from $O$ to the…

Let $M_n$ be the finite honeycomb-lattice region with four-colored cells, and let $B_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates between opposite sid…