19 problems

Detailed-balance classification conjecture. All non-trivial solutions to the detailed balance condition for $F_{\mathrm{AL},\mathrm{AL}}$ are given in Proposition (b).

Consider self-avoiding walk on ${\mathbb{Z}}^2$, and let its endpoint after $n$ steps be denoted by $S_n$. The endpoint scaling conjecture. The endpoint runs on scale … Determining…

Interior delocalization conjecture. If $(\lambda,h)\in\overset{\circ}{\mathcal D}$, then

Numerical delocalization conjecture. The points

Equality of the critical curves. The conjectures are

Let $\beta_N$ be the scaling defined in the paper, and let $chat\beta<\hat\beta_c$. Write $W_N(\beta_N)$ for the normalized partition function. The uniform moment-boundedness conje…

All-moments critical-point conjecture. For every $W>W_c$, the martingale $(\psi_n)_{n\geq 0}$ converges in $L^p$ for every $p>1$; equivalently,

Let $d\geq 3$, let $W_c(\mathbb{Z}^d)$ denote the critical point for the vertex reinforced jump process (VRJP) on $\mathbb{Z}^d$, and let $(\psi_n)_{n\geq 0}$ be its martingale. Th…

Derrida–Griffiths–Higgs conjecture. The following assertions hold: (i) the specific heat, namely the derivative of the finite-volume free energy, converges $\mathbb P$-almost surel…

Let $\mathpzc{HSLG}$ denote the half-space log-gamma polymer, let $P$ be a down-right path, and let $\theta$ and $\alpha$ be its model parameters. For each … let $z_{v,P}(\cdot)$ b…

Fix $t>0$, $\Delta>0$, two segments $[X,X']$ and $[Y,Y']$ in $\mathbb R$, and real numbers $(x_1,\dots,x_n)$, $(y_1,\dots,y_n)$, $(\tilde x_1,\dots,\tilde x_k)$, and…

The critical weak-disorder limiting regime for $(2+1)$-dimensional polymers studied by Caravenna, Sun, and Zygouras is expected to produce a continuum polymer model with an analogo…

Fix $k\geq 1$ and $c\in(0,\infty)$. Let $\alpha_1,\ldots,\alpha_k$ be real numbers and let $F(z_1,\ldots,z_k)$ satisfy $\alpha_k=0$, $\alpha_j>\alpha_{j+1}+c$ for $1\leq j\leq k-1$…

Mean-square displacement exponent conjecture. For dimension $1\leq d\leq 4$, there exists a strictly positive exponent $\gamma=\gamma(d,\alpha)$ and a constant $c$ such that

In a directed polymer model in a random environment, let $Z^{\beta}_m$ denote the total partition function for paths of length $m$ at inverse temperature $\beta$. Fluctuation expon…

In the directed i.i.d. case in dimension $d=2$, consider a point-to-point free energy in the KPZ universality class, centered and normalized according to its fluctuation scale. KPZ…

Van der Hofstad–Klenke conjecture. On the critical line $\gamma=\beta$,

Let $\alpha\in(0,\infty)$, let $\chi:=\sum_{n\in\mathbb{N}}[\textbf{P}(S_n=0)]^2$, and let $\beta_c$ denote the critical inverse temperature separating the annealed and quenched cr…

Let $mathcal{F}(T,X)$ denote the continuum directed random polymer free energy at time $T$ and spatial position $X$, and let ${mathscr{A}}2(X)$ be the ${rm Airy}2$ process. **Airy…