68 problems

In the strong disorder phase of a directed polymer in a random environment on ${{\mathbb Z}}^d$, trajectories are considered under the polymer measure in an environment that may co…

Let $W_n:=Z_n/\mathbf{E}(Z_n)$ be the normalized partition function, and let $W_n\to W_\infty$ almost surely. Define $\bar\beta_c\in[0,\infty]$ so that weak disorder holds for…

Consider disordered systems whose couplings are sampled from a probability distribution, including the Bernoulli distribution as a discrete example. Discrete-distribution conjectur…

Let $\mu_n$ be the polymer measure at time $n$, let $p(\beta)$ denote the quenched free energy, and let $\lambda(\beta)$ denote its annealed bound. Directed-polymer localization di…

Single-site approximation conjecture. There is a value $\lambda(\beta,\varepsilon,h_0)$ such that

High-temperature spectral-branch conjecture. Results analogous to those proved for the one-dimensional model should hold for sufficiently small $\beta$ in dimensions $d\geq 2$.

Consider directed copolymer and pinning models with disorder and tail exponent $\alpha=3/2$. Higher-order transition conjecture. The transition is of order higher than two, possibl…

Consider discrete disordered systems with an energy function generated by random disorder variables. Local REM universality conjecture. The local REM property should hold for discr…

Let the energies of a number-partitioning model be ordered increasingly, and consider the energy levels near a fixed energy. Local REM conjecture. The local energy spectrum should…

A disordered potential is a potential introducing spatial randomness into a lattice Schrödinger operator, and localization means that the operator has purely point spectrum with ei…

Imry–Ma conjecture. Adding a random magnetic field destroys the phase transition in dimension $2$, but not in dimensions $3$ and higher.

Let $\bm{u}(x)$ denote the polymer and let $\langle *\rangle$ denote its Gibbs measure. For points $x,y$ with large separation, consider the disorder-averaged squared displacement…

Let $\alpha>0$ be the exponent governing the inter-arrival law of the pinning renewal process, and let $p>0$ determine the environment correlation by … The critical point separates…

Incommensurate-lattice spectral-gap conjecture. The stronger result

Let $\gamma$ be the model parameter, let $\Delta_\gamma(H_2)$ denote the relevant boundary curve, and let $\alpha(\gamma)$ be the exponent defined by the preceding model equations.…

A disordered Ising ferromagnet on $\mathbb{Z}^2$ has Hamiltonian … where the coupling constants $\eta_{\{x,y\}}$ are i.i.d. positive random variables and $\sigma_x\in\{+1,-1\}$. A…

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 a one-dimensional disordered Klein–Gordon-type or disordered nonlinear Schrödinger-type chain be in the atomic limit, with an initially localized wave packet and finite initial…

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-moment growth conjecture. The critical moment satisfies

Crossover-scale conjecture. The crossover range is universally estimated by

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

In first-passage percolation, let $T_t(0,v)$ be the passage time from $0$ to $v$ at dynamical time $t$, and let $\pi_t(0,v)$ be a geodesic between these vertices at time $t$. The n…

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