117 problems

Sharp upper-bound conjecture. Without any further assumption on the model, the sharp upper bound for $p(\beta)$ should be the bound in equation (e117). The source compares this pro…

Let the wall attraction in the half-space directed random polymer vary between a strong attractive-force regime, also called the subcritical or bound phase, and a weaker-attraction…

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…

A generalized Hamilton–Jacobi polymer is a directed polymer model with a random potential whose path weights are governed by a generalized Hamilton–Jacobi structure. In this settin…

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…

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…

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…

RWRE–directed polymer correspondence conjecture. There should be a precise relation between the RWRE and directed-polymer models such that

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…

Assume $\beta_N$ as in the paper's critical scaling. The critical-point scaling-limit conjecture. At the critical point $\hat\beta=\hat\beta_c$, the properly rescaled partition fun…

Critical fluctuation conjecture. The random variables

For a directed polymer in a random environment, let $\beta_c$ denote the critical point separating weak and strong disorder, and let $\bar\beta_c$ denote the critical point separat…

Let $Z^{\beta_N}_N(0)$ denote the point-to-plane partition function of the directed polymer at the critical intermediate-disorder scale, and let $N$ tend to infinity. Critical one-…

Consider directed polymers in an independent and identically distributed random environment, with partition function formed from Gibbs weights involving an inverse-temperature para…

Let $f\in\mathcal C_c(\mathbb{R}^d)$, let $S_n(f)$ be the centered, spatially averaged directed-polymer partition function defined by … and let $p^*(\beta)$ denote the relevant mom…

For $u<0$, let $\mathbb{Q}^{u,t}_0$ be the point-to-stationary-measure half-space continuum directed random polymer measure, let $X_t$ be its endpoint displacement, and let…

Fix $u eq 0$ and let $Hu(t,0)$ denote the stationary half-space KPZ height at time $t$ and position $0$. Gaussian fluctuation conjecture. As $t$ tends to infinity, one should h…

Let $N$ be fixed, let $\gamma$ denote the parameters of the coupled Markov chain, and consider the coupled chain defined by, its dual chain defined by, the system of SDEs, and its…

Let $\kappa_n(t)$ be the quenched $n$-th cumulant of the endpoint of the point-to-line directed polymer, and let $R_\beta^{(\theta)}$ be the covariance function defined in the pape…

Let $\Sigma_L(\beta)$ denote the effective diffusivity of the winding number of a directed polymer in a white noise environment with period $L$, and let $\Sigma_1(\beta)$ be the di…

Infinite-order criticality conjecture. The free energy satisfies

Critical-moment growth conjecture. The critical moment satisfies

Let the two-dimensional critical stochastic heat flow be the flow of random measures on $\mathbb R^2$ constructed as the scaling limit of intermediate-disorder directed-polymer par…

Let a height process arise from a polymer model with a multilinear chaos expansion in an underlying environment field, and let $F$ be a bounded measurable functional of the height…

Let $X_n$ denote the endpoint of the polymer at time $n$, and let $\langle\cdot\rangle$ denote expectation under the Gibbs measure. The annealed variance is the expectation over th…