24 problems

Let $D_{\sin}(s,\gamma)=\det(\mathcal I-\gamma\mathcal K_{\sin})$ be the sine-kernel Fredholm determinant, where $\mathcal K_{\sin}$ acts on $L^2(-s,s)$ with kernel … Here…

Let $D_n$ be the normalized partition function defined by … and let $L_n$ and $\Gamma_0$ be the corresponding quantities for the limiting entropy-regularized optimal-transport prob…

Let $\lambda=0$, so that $q=-1$, and consider Fredholm determinants of the type defined by the source's equation for $\operatorname{Frdet}$. The relevant integral equations are the…

Let $q$ be the deformation parameter, $t$ the time parameter, and let $\mathcal{D}$ denote the discrete $q$-lattice … Let $\mathcal{D}$ be the matrix Fredholm determinant defined e…

Let $J$ be a jump matrix with factorization $J=\Phi_-^{-1}\Phi_+$, let $\tau_W[J]$ be the Widom determinant, let $\tau_{AB}[J]$ be the corresponding Arinkin–Borodin tau function, a…

Main conjecture. Under the spectral-gap and Lipschitz-eigenfunction assumptions, the limit defining the normalized partition function holds with the Fredholm determinant…

Fix $r_1,\dots,r_n\in\mathbb{Z}$, $x_1,\dots,x_n\in\mathbb{R}$, $t>0$, and an initial height function $h$. Let $K_r$ be the kernel defined using these parameters, and let…

Kitaev's determinant conjecture. Under these assumptions, the Fredholm determinant satisfies

Let $H$ be a group, let $h\in H$, and let $D_P$ and $D_{\mathcal P_M}$ be the operators associated with the boundary conditions $P$ and ${\mathcal P}_M$. Let $A$ be the relevant bo…

Let $A$ be the function defining the Zakharov–Shabat reflection coefficient, let … and let $A_s$ be the associated operator, with kernel $K_s=A_s^2$. Consider the Riemann–Hilbert p…

Fermion–Marino conjecture. The Hamiltonian system implies that $q_0$ solves the equation of order $2n$ in the Painlevé II hierarchy with

The beta model Fredholm determinant conjecture. For the beta model,

The q-Hahn PushTASEP Fredholm determinant conjecture. For the step initial configuration,

The one-point height distributions and Fredholm-determinant expressions obtained for the described crossover initial conditions are conjectured to be universal for all models in th…

Large-negative-$t$ asymptotics conjecture. As $t\to-\infty$,

Let $q$ satisfy $0<q<1$, let $x_n(t)$ denote the position of particle $n$ at time $t$, and define … For all $\zeta\in\mathbb{C}\setminus\mathbb{R}_{>0}$, the conjectural Fredholm d…

Painlevé II determinant constant conjecture. The undetermined constant satisfies

Let … denote its $\lambda$-generalisation. In the discrete Fredholm determinant representation of … . This conjecture concerns the extension of the Ising-model form-factor expansio…

Assume that $0\leq\operatorname{Re}(z+z')<1$ and $z,z',w,z+z'+w\notin\mathbb{Z}$, as in Proposition. Let $C_L$ be the constant in the small-$s$ asymptotics of the confluent hyperge…

Assume the hypotheses of Proposition. Let $C$ be the constant in the asymptotic expansions of the hypergeometric-kernel determinant $D(t)$ as $t\to1$, and define … where $G$ is the…

For vanishing magnetic field, let $\tau_{\scriptscriptstyle PBT}(s)$ denote the relevant Painlevé correlation function, and consider the theory of free massive Dirac fermions on th…

Let $\mu(\tau)$ denote the parameter-dependent quantity in the interpolating random-matrix ensemble, and let the even and odd terms refer to products formed from the even- and odd-…

Let $I+V$ be the generalized sine-kernel operator (GSK), let $x$ be the large asymptotic parameter, and let $\nu$ denote the parameter whose integer shifts restore the determinant'…

Let $p$ and $q$ be the ASEP jump parameters with $p<q$, let $x_m(t)$ denote the position of the $m$-th particle at time $t$, and let $K_0$ be the operator on $L^2(\mathbb {R})$ wit…