Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.35681

Power variations of critical Gaussian multiplicative chaos and their spectral applications

Mo Dick Wong
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce power variations of critical Gaussian multiplicative chaos (GMC) along refining partitions in arbitrary dimension. Under suitable renormalisation, we prove uniform fractional moment bounds via Laplace transform estimates as well as stable convergence to supercritical GMCs. Together, these results provide a unified approach to regularity and spectral questions for critical chaos, yielding: (i) the sharp exponent for the logarithmic modulus of continuity, (ii) the existence of non-empty essential spectrum for critical Liouville quantum gravity surfaces; and (iii) an almost-sure quantitative Fourier decay.

Explore related subjects

math.PRmath-phmath.FAmath.MPmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mo Dick Wong. 2026-09-28. Power variations of critical Gaussian multiplicative chaos and their spectral applications. https://arxiv.org/abs/2609.35681

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rough backward SDEs with discontinuous Young drivers

We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$.

math.PR↗
arXiv2026

Convergence of the KMP model to the KPZ equation

We prove that the Kipnis-Marchioro-Presutti (KMP) process converges to the Kardar-Parisi-Zhang (KPZ) equation, as time $t$ goes to infinity, in a properly scaled observation window shifted by $t^{3/4}$. Our proof is based on identifying the KMP process with a stochastic flow of kernels describing transition probabilities in a certain model of random walk in space-time random environment. This allows to apply a recent result of arXiv:2401.06073 proving convergence of the density field of random walks in random environment to the KPZ equation in a suitably general sense.

math.PR↗
arXiv2026

Permuton and local limits for the Luce model

We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the weights, we establish a permuton limit theorem describing the global behavior of Luce-distributed permutations and derive an explicit density of the limiting permuton. We further compute limiting pattern densities and analyze the differences between exact Luce permutations and their permuton approximations. We also study the local convergence of these permutations, proving a quenched Benjamini--Schramm limit and a central limit theorem for consecutive pattern occurrences. Finally, we prove a central limit theorem for the number of inversions.

math.PR↗