20 problems

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…

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…

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

Let $\mathcal{Z}(s,x;t,y)$ denote the continuum directed random polymer partition function, and define … Let $\mathcal{S}(x,y)$ be the parabolic Airy sheet. KPZ sheet-to-Airy-sheet…

Fix $t\in\mathbb Z$. In the half-space Beta random walk in random environment, define … for $x\in\mathbb Z_{\geqslant 0}$. Half-space Gamma-distribution conjecture. The limit exist…

Let $t\in\mathbb Z$ and consider the full-space Beta random walk in random environment. For each $x\in\mathbb Z$, let $\mathsf Z_t^{\mathbb Z}(x)$ denote the full-space point-to-li…

Weak-disorder moment conjecture. For all $\delta\in(0,1)$ and $\hat\beta\in(0,1)$, uniformly for all $q<(1-\delta)q_\star(N,\hat\beta)$, as $N\to\infty$,

Let $F_q(\beta)$ and $F_a(\beta)$ denote the quenched and annealed free energies of the directed polymer, respectively, let $d_s$ be the spectral dimension of the underlying fracta…

Let the underlying random walk satisfy the local limit theorem in the sense discussed by Croydon and Hambly. For $T>0$, let $Z^0_{Tn,\beta_n}$ be the directed-polymer partition fun…

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

For $c$, $\beta$, and \frac{1}{N^2}\frac{1}{2}} as in the preceding definitions, let $R^-_c(]\alpha,\beta)$ and $R^+_c(]\alpha,\beta)$ denote the lower and upper limits of the scal…

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 the Hamiltonian be convex and let its associated Lagrangian have superlinear growth. A generalized directed polymer is the polymer process associated with the corresponding ran…

Let $R_5:=\{(\gamma_c,\gamma_a): \gamma_a>1/2,\ \max\{0,2/\gamma_a-1,(\gamma_a-5)/(\gamma_a-2)\}<\gamma_c<3/(2\gamma_a)\}$. Consider the directed-polymer partition function…

Assume (Dir), (Ind), (Loc), $V\in\mathcal{L}$, and $\Lambda_q(V)<\infty$. Let $\underline h_\infty^\lambda$ and $\bar h_\infty^\lambda$ denote the lower and upper limiting free-ene…

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 $\omega$ be an i.i.d. random environment. For each $\beta>0$, let $Z_n^{\omega}(0;\beta)$ be the directed polymer partition function, and let $F_{GUE}$ denote the Tracy–Widom G…