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Sparse autoencoders (SAEs) decompose transformer residual streams into interpretable feature dictionaries, yet the relationship between SAE width and causal influence on model output has not been systematically characterised. We introduce…

We study the convolution of functions of the form \[ f_\alpha (z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^\alpha - 1}{2 \alpha}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $\alpha\in(0,1)$. We…

We prove that quadrangulations with a simple boundary converge to the Brownian disk. More precisely, we fix a sequence $(p_n)$ of even positive integers with $p_n\sim 2\alpha \sqrt{2n}$ for some $\alpha\in(0,\infty)$. Then, for the…

We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe…

We prove that a uniform rooted plane map with n edges converges in distribution after a suitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambj{\o}rn and Budd allows to derive this result…

We study the potential theory of a large class of infinite dimensional L\'evy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e. excessive functions with…

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family ($\mathrm{BD}_L$, $0 < L < \infty$) of random metric spaces homeomorphic to the closed…

We show that, for $\kappa\in(0,8)$, the integral of the laws of two-sided radial SLE$_\kappa$ curves through different interior points against a measure with SLE$_\kappa$ Green function density is the law of a chordal SLE$_\kappa$ curve,…

Once recognizing that point particles moving inside the extended version of the rippled billiard perform L\'evy flights characterized by a L\'evy-type distribution $P(\ell)\sim \ell^{-(1+\alpha)}$ with $\alpha=1$, we derive a generalized…

A classical theorem of Colin de Verdi\`ere shows that on a closed manifold of fixed topology one can prescribe an arbitrary finite portion of the Laplace-Beltrami spectrum (including multiplicities, subject to the usual topological…

We construct an aggregation process of chordal SLE(\kappa) excursions in the unit disk, starting from the boundary, growing towards all inner points simultaneously, invariant under all conformal self-maps of the disk. We prove that this…

We study the shape of the outer envelope of a branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$. We focus on the extremal particles: those whose norm is within $O(1)$ of the maximal norm amongst the particles alive at time $t$.…

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by…

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally…

We explore the statistical behavior of the order statistics of the flights of One-sided Levy Processes (OLPs). We begin with the study of the extreme flights of general OLPs,and then focus on the class of selfsimilar processes,investigating…

This paper examines how close the chordal $\SLE_\kappa$ curve gets to the real line asymptotically far away from its starting point. In particular, when $\kappa\in(0,4)$, it is shown that if $\beta>\beta_\kappa:=1/(8/\kappa-2)$, then the…

We define radial exploration processes from $a$ to $b$ and from $b$ to $a$ in a domain $D$ of hexagons where $a$ is a boundary point and $b$ is an interior point. We prove the reversibility: the time-reversal of the process from $b$ to $a$…

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to…

Let A_2 be the moduli stack of principally polarized abelian surfaces and V a smooth l-adic sheaf on A_2 associated to an irreducible rational finite dimensional representation of Sp(4). We give an explicit expression for the cohomology of…

This paper introduces the annulus SLE$_\kappa$ processes in doubly connected domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a time-change. For $\kappa\not=6$, some weak equivalence relation exists between annulus…