26 problems

For a Fisher–Hartwig determinant of size $N$, the relevant singularities are expected to produce additional logarithmic terms in its asymptotic expansion. Fisher–Hartwig's conjectu…

Let $R_n(t)$ be defined by … Then … The largest eigenvalue, or zero-momentum occupation number, is … Here $A=e^{\frac{1}{12}}e^{-\zeta'(-1)}$ is Glaisher's constant. Dyson's conjec…

Basor–Tracy conjecture. As $n\to\infty$,

Let $\phi$ be a symbol on the unit circle satisfying the assumptions of the strong Szegő limit theorem, and let $z_1,\ldots,z_N$ be distinct points on the unit circle. For paramete…

Generalized Fisher–Hartwig conjecture. The determinant ratio satisfies

Let $\{f_k\}_{k=0}^{n}$ be a finite Pólya frequency sequence of order $d\geq r$, and let $P_n^r(z)$ be the higher-order Toeplitz determinant polynomial defined in the preceding con…

Let $\{f_k\}_{k=0}^{n}$ be a finite numerical sequence and, for $r\geq 2$, define … Here $(z)_k$ is the Pochhammer symbol. Karp's higher-order conjecture. Suppose the polynomial…

Let $\{f_k\}_{k=0}^{n}$ be a finite numerical sequence, and define … where $(z)_0=1$ and $(z)_k=z(z+1)\cdots(z+k-1)$ for $k\geq 1$. Karp's conjecture. Suppose the polynomial…

For $N>1$, consider the symbol … Let $x_n^{(N)}$ denote the associated quantity, and let $J_n^{(N)}(\xi_1,\dots,\xi_N)$ denote the generalized Bessel function appearing in the pape…

Let $T_\ell(\mathcal M)$ be a block Toeplitz matrix with a $2\times2$ matrix symbol $\mathcal M(p)$, whose determinant is nonzero and has zero winding number, and suppose that…

Let $f^{(\nu)}$ be the symbol defined on $[-\pi,\pi)$ by … with $\nu\geq0$, and let $s_k^{(\nu)}$ denote the Fourier coefficients of $\log f^{(\nu)}$, with $s_0^{(\nu)}$ its zeroth…

Ares–Fisher conjecture. The logarithmic coefficient associated with the discontinuity at $\theta_r$ is

Even-interval vanishing-order conjecture. In the limit $\epsilon\to 0$,

Conjectured vanishing-order formula. For all $n\geq 1$,

Let $\chi(\kappa,k_F,L)$ be the generating function with Fisher–Hartwig expansion … whose two parts are given by equations (1) and (2). The parameter $\kappa$ may have half-integer…

Let $f(phi)$ be a smooth function, and consider the determinant whose asymptotic expansion is prescribed by the terms in the previously described expansion, with each subleading br…

Let $PF_r$ be the class of sequences whose infinite Toeplitz matrix has all minors of order at most $r$ non-negative. For $r\geq2$, define … Here $(x)_k$ is the rising factorial, a…

Let $\{f_k\}_{k=0}^{n}$ be a Pólya frequency sequence of infinite order, denoted $PF_{\infty}$, meaning that all minors of the associated infinite Toeplitz matrix are non-negative.…

Let $\{f_k\}_{k=0}^{n}$ be a non-negative sequence, let $\alpha,\beta>0$, and define … Here $(x)_k$ denotes the rising factorial. Karp–Sitnik's coefficient-positivity conjecture. I…

Let $\chi(\kappa,x)$ have the asymptotic expansion … where $L_j(\kappa,x)$ are Laurent series in $x$. All-orders periodicity conjecture. For every order in the $1/x$ expansion, the…

Let $\chi(\kappa,x)$ have the Fisher–Hartwig asymptotic expansion given by Eqs. (GFH-1)–(GFH-2), and let the coefficients be defined for values of $\kappa$ away from the lines…

Let $\chi(\kappa,x)$ be the double-scaling limit of the Toeplitz determinant, with asymptotic expansion … where $L_j(\kappa,x)$ are Laurent series in $x$ whose coefficients depend…

Let $D_N$ be a Toeplitz determinant with Fisher–Hartwig branches, and let $N$ tend to infinity. For each branch, consider its asymptotic expansion in powers of $1/N$. Full-branch c…

Let $D_N$ be a Toeplitz determinant whose asymptotics are described by multiple Fisher–Hartwig branches, and let $N$ tend to infinity. Subleading-branch conjecture. Subleading Fish…

Let $F_1(k_F,\lambda)$ denote the coefficient of $L^{-1}$ in the expansion … for the full-counting-statistics generating function of one-dimensional free fermions. The cubic coeffi…