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Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. Popular clustering algorithms such as Lloyd's algorithm and $k$-means++ can take $\Omega(ndk)$ time when clustering $n$ points in…

Probabilistic clustering models (or equivalently, mixture models) are basic building blocks in countless statistical models and involve latent random variables over discrete spaces. For these models, posterior inference methods can be…

We study the fractal geometry of O($n$) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable…

We establish densification converses for Walker LEO constellations under nearest-visible association in the full-frequency-reuse setting. Performance is evaluated under the invariant (stationary) measure induced by the constellation/Earth…

We study behaviour of the critical $O(N)$ vector model with quartic interaction in $2 \leq d \leq 6$ dimensions to the next-to-leading order in the large-$N$ expansion. We derive and perform consistency checks that provide an evidence for…

We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$.…

A general two-dimensional spin model with U$(N)$ invariance, interpolating between $\CPN$ and ${\rm O}(2N)$ models, is studied in detail in order to illustrate both the general features of the $1/N$ expansion on the lattice and the specific…

Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the $O(n)$ model (symmetry group $O(n)$) and the Potts…

We present the perturbative results of the discretization errors proportional to the quark mass ($\mathcal{O}(a m)$) on the QCD running coupling within lattice perturbation theory. Our analysis involves calculating the 2-loop…

We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical $ O(N) $ model, to fourth order in the $ \epsilon $-expansion. This is done fully…

Deep generative models aim to learn the underlying distribution of data and generate new ones. Despite the diversity of generative models and their high-quality generation performance in practice, most of them lack rigorous theoretical…

We show that the large $n$ limit of the $O(n)$ quantum rotor model defined on a general graph has the same critical behavior as the corresponding quantum spherical model and that the critical exponents depend solely on the spectral…

We numerically study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model by using the transfer matrix and the worm Monte Carlo methods. The renormalization group flow is extracted from the landscape…

We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by…

This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general…

We study the branching tree of the perimeters of the nested loops in critical $O(n)$ model for $n \in (0,2)$ on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade…

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q -> 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have…

We present an exact solution of the $O(n)$ model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed…

We show how to compute the probability distributions of the order parameter of the $O(n)$ model at two-loop order of perturbation theory generalizing the methods developed for computing the same in case of the Ising model…

We investigate the O($n$) nonintersecting loop model on the square lattice under the constraint that the loops consist of ninety-degree bends only. The model is governed by the loop weight $n$, a weight $x$ for each vertex of the lattice…