24 problems

Uniform-integrability conjecture. Under this condition, the family $\{\tilde Z_N\}_{N\in\mathbb N}$ is uniformly integrable and converges weakly to an $L^1$-integrable limit.

Let $H\in(\frac12,1)$ be the correlation exponent, let $\alpha$ be the tail exponent of the renewal process, and let $\tilde Z_{N,\beta_N}^{\omega,h_N}$ denote the rescaled Wick-or…

Critical-height concentration conjecture. When $h=h_w$, the surface height should concentrate around $H_w$, with fluctuations similar to those in the subcritical regime described b…

Infinite-order transition conjecture. The phase transition is of infinite order for every $\alpha\in(0,1)$.

Weak-interaction scaling conjecture. In all three regimes,

Let $\tau$ and $\hat\tau$ be renewal sets with tail exponents … , respectively, and let $h_c(\beta)$ and $h_c^a(\beta)$ denote the quenched and annealed critical points of the pinn…

Stable-noise scaling-limit conjecture. The sequence

Let $F(\beta,h)$ be the quenched free energy, let $h_c(\beta)$ be the critical point, and suppose the renewal inter-arrival law satisfies the regular-variation assumption and the e…

Let $\tau$ be the renewal process with inter-arrival law $K$, and let $h_c(\beta)$ denote the critical point at disorder intensity $\beta$. Assume that the environment satisfies th…

Let $\rho$ denote the decay exponent of the collision probability ${{\mathbf P}}_{X,Y}(X_n=Y_n)$ in the random walk pinning model, and let $\beta_c^{\mathrm{que}}$ and…

Mixing-time conjecture. For every $\lambda>0$ and $d>1$, the mixing time is bounded by

Let $\hat{\beta}>0$ be finite, and let $\hat{I}_{{\hat{\beta}},\infty}$ be the infinite-volume variational set associated with the model. Finite-point conjecture. We conjecture tha…

Let $\gamma=0$, and let the renewal law satisfy $K(n):=\mathrm{P}(\tau_1=n)\sim L(n)n^{-\rho}$ with $\rho>1$. Let $\mathrm{P}_{{\beta}_N,h,N}^{{\omega}}$ denote the corresponding p…

Let $\tau$ be the renewal process and let $N$ be the system size, so that $\tau/N$ denotes its rescaled renewal set. In the regime $\gamma\geq 1$, the entropy optimization yields t…

Disorder irrelevance conjecture. For fixed $\beta>0$ sufficiently small, the limit in distribution of ${\mathrm P}_{NT,\beta,h_N}^{\omega,\rm c}(\mathrm{d}(\tau/N))$ as…

Let $h_c(\beta)$ be the critical curve of a disordered pinning model, and let $\boldsymbol h_c^{(\alpha)}(1)$ be the critical curve of the continuum model at unit coupling. For a s…

The sharp general criterion. One has

The conjecture. The assumption that the correlations between the events $\mathfrak{F}_{\mathcal{E}}$ are non-increasing is not necessary for the criterion

Consider the field of pinned sites in the disordered pinning model in dimension $d\geq 3$. Let the pinning strength tend to zero, and apply a suitable rescaling to the pinned-site…

Let $(\varphi_x)$ be the interface field of the disordered pinning model on $\mathbb{Z}^d$, with the Gaussian nearest-neighbor interaction $(\varphi_x-\varphi_y)^2$ replaced by a u…

Let $B$ and $\kappa$ satisfy … Consider the annealed hierarchical pinning model and its corresponding pure model. Smoother phase transition conjecture. The phase transition of the…

Cheliotis–den Hollander conjecture. For every $\mu_0$,

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…

Derrida–Hakim–Vannimenus disorder relevance conjecture. Disorder is relevant when $1/2<lpha<1$, irrelevant when $lpha<1/2$, and marginally relevant when $lpha=1/2$. More precisely,…