28 problems

Let $\mathsf{L}_N$ be the C$\beta$E field on $\mathbb{T}$, with parameter $\beta>0$, and suppose the critical relation $\beta=2s^2$ holds. Let $\mathfrak{G}_\beta$ denote the expli…

FHK postulate. The interval counts $\delta\mathcal{N}_I^{({\boldsymbol J})}$ should be log-correlated and, as $\Delta\to\infty$, should converge to a Gaussian random model of the f…

Let $P_N(z)$ be the characteristic polynomial of an $N\times N$ random matrix, and let $\mathcal{I}$ be a fixed macroscopic interval in the bulk of the spectrum or a fixed finite a…

Let $D\subset\mathbb{R}^d$ and let a log-correlated Gaussian field on $D$ admit a smooth white noise decomposition. Denote its derivative Gaussian multiplicative chaos by…

Let $D_n(\cdot,\cdot)$ be the exponential metric, normalized by the median of the set-to-set distance, and suppose that a limiting metric exists in the phase where…

A log-correlated field is a random field with logarithmically divergent short-range correlations. For a regularisation parameter tending to zero, consider the field's maximum after…

Log-correlated fields are random fields whose correlations have a logarithmic singularity at short distances. The local extrema and maximum of such a field may exhibit universal li…

GMC universality conjecture. The result should correctly describe the behaviour of analogous moments of moments of random matrices, or more generally of structures which are asympt…

For $x\in\mathbb{Z}^2$ and $N\geq 1$, let $\ell_x^N$ be the total number of visits to $x$ by a planar simple random walk started at the origin before exiting $[-N,N]^2$. Define the…

Log-correlated Gaussian fields are random fields whose covariance between values at two different sites decays logarithmically with their distance; examples include the two-dimensi…

A field with logarithmic or approximate logarithmic correlations is a random field whose correlations decay logarithmically, including hierarchical fields as an example. Universal…

Branching Brownian motion is one model with scale-free correlations, and related models include branching random walk and the two-dimensional Gaussian free field. The paper establi…

Fyodorov–Simm conjecture. As $N\to\infty$,

Universal maximum conjecture. The maximum satisfies, for some constant $C$ and a random variable $Z$ called the derivative martingale,

Atomic-limit conjecture. These sums converge in law to an atomic random measure $\nu$.

Triple-point conjecture. As $\varepsilon\to0$,

Supercritical renormalization conjecture. As $varepsilon\to0$,

Let $(X_n)_n$ be a cutoff approximation of a centered Gaussian field on $[0,1]^d$ with logarithmic covariance, and set $c_n={\rm Var}(X_n)$. Let $M'$ be the derivative chaos and le…

The frozen-phase random measures $N'_alpha$ are constructed from the critical measure $M'$ by a stable point process, with $0<alpha<1$. Freezing conjecture. These families of measu…

A logarithmically correlated model is a random model whose correlations decay logarithmically with distance; its maximum is the largest value attained by the model. Universality co…

Let $g_\beta(y)$ be the generating function associated with a logarithmic random landscape at inverse temperature $\beta$, and let the critical inverse temperature be $\beta_c=1$.…

Consider the extreme-value/free-energy distribution of a logarithmically correlated process, with $c$ denoting the coefficient appearing in the model's rescaling formula. For short…

Let $p(x)$ denote the probability density of the properly centered and rescaled maximum of a logarithmically correlated random function. The backward tail concerns the regime…

Carpentier–Le Doussal conjecture. The maximum has the asymptotic form

Maximum-law conjecture. There is a constant $c>0$ and a limiting random variable $G_d$ such that $Y_t$ converges in law to $G_d$ as $t\to\infty$, with