14 problems

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…

Critical-moment growth conjecture. The critical moment satisfies

Strict inequality conjecture. There exists no $\beta\in\mathbb R_+$ such that

Quenched localization conjecture. There exists an environment-measurable process $(\Gamma_t)_{t\geq 0}$ with $\Gamma_t\subseteq\omega$ and

Entropy criterion conjecture. The weak-disorder critical point is finite if and only if

Let $u$ be the law of the random environment, and let $\beta$ denote the inverse temperature. For a polymer measure $mu{\beta,n}$ on paths of length $n$, let $ell(mu{\beta,n})$ and…

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 $\beta=\sigma+i\tau\in\mathbb C$, let $\rho\in[-1,1]$, and let $p_t(\beta)$ be the log-partition function per unit time for the complex BBM energy model. Let $B_1,B_2,B_3$ deno…

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…

Let $\beta>0$ denote the disorder strength, let $F(\beta)$ be the directed-polymer free energy, and let $\boldsymbol F(1)$ be the corresponding continuum free energy at unit coupli…

Let $h_c^\mathrm{que}(\beta)$ be the quenched critical curve and let … when this limit exists, for a copolymer model whose renewal exponent is $\alpha>1$. Critical-slope conjecture…

Path-localization conjecture. The set of inverse temperatures for which path localization holds is

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