258 problems

Let $M_n$ be drawn from the fixed-trace Gaussian Unitary Ensemble, and let $\lambda_1^n<\cdots<\lambda_n^n$ be the eigenvalues of $M_n/\sqrt n$. Define the centered empirical measu…

Let $n(r)$ be the number of zeros of the planar Gaussian analytic function $\mathbf g$ in the disk $D(0,r)$. Sodin's deviation conjecture. As $r\to\infty$, … This predicts the loga…

Let $n(r)$ denote the number of zeros of a planar or hyperbolic Gaussian analytic function in the disk of radius $r$ centered at the origin, with $r<1$ in the hyperbolic case. Pere…

Let $\chi_\mathrm{orb}$ denote orbital free entropy and let $i^*$ denote free mutual information for tuples of subalgebras in a tracial $W^*$-probability space. Orbital free entrop…

Let $T\to\infty$ and let $k>0$. Consider $t$ uniformly in $[T,2T]$, and let $a_k$ be the arithmetic factor from the zeta-function moment conjecture and…

Let $T_{n,p}$ be the number of triangles in an Erdős–Rényi random graph $G(n,p)$. For fixed $[?]$ and as $n\to\infty$ and $p\to 0$ slower than $n^{-1/2}$, consider the upper-tail p…

Let $\rho$ and $\sigma$ be quantum states in the state space $\mathcal{S}$. The rate functions $\mathcal{I}^c_{\operatorname{Id}}(\rho,\sigma)$ and $\hat{I}(\rho,\sigma)$ are defin…

Integrability-condition conjecture. The condition $\mathbf{E}Z_{1}\log Z_{1}<\infty$ can be dropped in all of the paper's theorems.

Let $\hat f$ be the Viana map described above, and let $\mathcal U$ and $\widetilde{\mathcal U}$ be the corresponding neighborhoods of $\hat f$. For each map…

Greven–den Hollander conjecture. For every $x>v$,

Consider constrained homogeneous random walks in ${\cal Z}_+^d$ and the associated problems of computing stationary probability distributions and large-deviation rates. The source…

Boundary-contribution conjecture. Since the Jensen–Varadhan functional can be interpreted as the bulk limit of the WASEP large-deviation functional, equations (LD boundary 1) and (…

Let $p$ be fixed, let $J$ denote the disorder law, let $V$ be the relevant energy parameter, and let $\mathscr C_N^J(V)$ be the annealed critical-point complexity. Assume that the…

Let $A$ be the random matrix considered above, and let $k$ denote the corank parameter, so that the event in question is $\operatorname{rank}A\le n-k$. The elementary event that…

Let the geometric $\alpha$-stable process be the symmetric Lévy process with characteristic exponent $\Psi(\xi)=\log(1+|\xi|^\alpha)$, and let $d$ denote the spatial dimension. The…

Let $S^{\mathrm{Keyl}}$ denote Keyl's rate function, and let admissible tomographic rate functions be rate functions arising from consistent tomography protocols. The Keyl conjectu…

Let $k\geq 2$ be an integer, let $\chi$ be a real non-principal Dirichlet character modulo $q$, and let $f=\mu^{(k)}\chi$ or $f=\mu^{(k)}g_\chi$. Then the large-values limsup conje…

Let $\nu_k([V,\infty))$ denote the upper tail of the limiting distribution associated with the summatory function considered in the paper. Meng's Large Deviation Conjecture. The ri…

Let $M(x)=\sum_{n\leq x}\mu(n)$ denote the summatory Möbius function. Ng's large-deviation conjecture. There exists a constant $B>0$ such that … This conjecture concerns the precis…

Let $d=2$, let $X_n$ be a random walk in an i.i.d. uniformly elliptic random environment, and let $I_q:\mathbb R^2\to[0,\infty]$ be its quenched large deviation rate function. Writ…

Let $X_n$ be a random walk in an i.i.d. uniformly elliptic random environment on $\mathbb Z^2$, and let $P_{0,\omega}$ denote its quenched law. Under the specific scaling appropria…

Beta-one conjecture. Theorem may apply for a large range of probabilities when $\beta_{m,n}$ is replaced by $1$.

Let $(a_k)_{k\geq 1}$ be a lacunary sequence satisfying … where $\eta$ is transcendental. Let $S_n$ be the associated lacunary trigonometric sum, and let the corresponding independ…

Consider the ground-state energy of an elastic manifold with internal dimension $0<d<4$ in the massless continuous limit. Let $\xi<2$ be the exponent governing the typical fluctuat…

Let $(X,\Delta)$ be a log Fano manifold admitting a Kähler–Einstein metric. Let $Z_{\boldsymbol{m}_{N}}$ be the K-reduced partition function, let $\mathcal{M}$ be the Mabuchi funct…