arXiv subjects

Guillaume Valette

Publications and source records attributed to Guillaume Valette.

At least 19 recordsLinked to original sources

arXiv2026

Sobolev embedding theorem and subanalytic measures

We focus on Borel measures that have a globally subanalytic density function. We prove, given such a measure $μ$ on a set $A$ and a globally subanalytic mapping $Φ:A\to Ω$, with $Ω$ bounded open subset of $\mathbb{R}^n$, a Sobolev embedding theorem for the Sobolev space $W^{k,p}_{Φ_*μ}(Ω)$ of the push-forward measure $Φ_*μ$. We derive an embedding of $W^{k,p}_{Φ_*μ}(Ω)$ into the space of inner Lipschitz functions and give an application to kernel theory.

math.AG↗

arXiv2026

Inner Lipschitz approximation in o-minimal structures

Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems.

math.AG↗

arXiv2026

Density of Neumann regular smooth functions in Sobolev spaces of subanalytic manifolds

We give characterizations of the bounded subanalytic $\mathscr{C}^\infty$ submanifolds $M$ of $\mathbb{R}^n$ for which the space of Neumann regular functions is dense in Sobolev spaces. By ``Neumann regular function'', we mean a function which is smooth at almost every boundary point and whose gradient is tangent to the boundary. In the case $p\in [1,2]$, we prove that the Neumann regular elements of $\mathscr{C}^\infty(\overline{M})$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at almost every boundary point. In the case $p$ large, we show that the Neumann regular Lipschitz elements of $\mathscr{C}^\infty(M)$ are dense in $W^{1,p}(M)$ if and only if $M$ is connected at every boundary point. The proof involves the construction of Lipschitz Neumann regular partitions of unity, which is of independent interest.

math.AP↗

arXiv2025

On subanalytic geometry

These notes constitute a survey on the geometric properties of globally subanalytic sets. We start with their definition and some fundamental results such as Gabrielov's Complement Theorem or existence of cell decompositions. We then give the main basic tools of subanalytic geometry, such as Curve Selection Lemma, Lojasiewicz's inequalities, existence of tubular neighborhood, Tamm's theorem (definability of regular points), or existence of regular stratifications (Whitney or Verdier). We then present the developments of Lipschitz geometry obtained by various authors during the four last decades, giving a proof of existence of metric triangulations, introduced by the author of these notes, definable bi-Lipschitz triviality, Lipschitz conic structure, as well as invariance of the link under definable bi-Lipschitz mappings. The last chapter is devoted to geometric integration theory, studying the Hausdorff measure of globally subanalytic sets, integrals of subanalytic functions, as well as the density of subanalytic sets (the Lelong number) and Stokes' formula.

math.AG↗

arXiv2025

$W^{1,p}$ priori estimates for solutions of linear elliptic PDEs on subanalytic domains

We prove a priori estimates for solutions of order $2$ linear elliptic PDEs in divergence form on subanalytic domains. More precisely, we study the solutions of a strongly elliptic equation $Lu=f$, with $f\in L^2(\mathcalΩ)$ and $Lu=div (A(x) \nabla u)$, and, given a bounded subanalytic domain $\mathcalΩ$, possibly admitting non metrically conical singularities within its boundary, we provide explicit conditions on the tangent cone of the singularities of the boundary which ensure that $||u||_{ W^{1,p}(\mathcalΩ)}\le C||f||_{L^2(\mathcalΩ)}$, for some $p>2$. The number $p$ depends on the geometry of the singularities of $δ\mathcalΩ$, but not on $u$.

math.AP↗

arXiv2024

$L^p$ Hodge theory for bounded subanalytic manifolds

Given a bounded subanalytic submanifold of $\mathbb{R}^n$, possibly admitting singularities within its closure, we study the cohomology of $L^p$ differential forms having an $L^p$ exterior differential (in the sense of currents) and satisfying Dirichlet or Neumann condition. We show an $L^p$ Hodge decomposition theorem, an $L^p$ de Rham theorem, as well as a Lefschetz duality theorem between $L^p$ and $L^{p'}$ forms (with $\frac{1}{p}+\frac{1}{p'}=1$) in the case where $p$ is large or close to $1$. This is achieved by proving that de Rham's pairing between complementary $L^p$ differential forms induces a pairing between cohomology classes which is nondegenerate (for such $p$). The main difficulty to carry it out is to show the density (in Sobolev spaces of differential forms) of forms that vanish near some singularities and are smooth up to the closure of the underlying manifold in the case $p$ large (Theorem 4.1). This result, which is of independent interest, also makes it possible to give a trace theorem that leads us to some explicit characterizations of Dirichlet and Neumann conditions in terms of traces and residues.

math.AG↗

arXiv2024

On Sobolev spaces of bounded subanalytic manifolds

We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$. We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator) for all $p\le p_M$, where $p_M$ is the codimension in $M$ of the singular locus of $\overline{M}\setminus M$. In the case where $M$ is normal, i.e. when $B(x_0,\varepsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\varepsilon>0$ small, we show that $\mathscr{C}^\infty(\overline{M})$ is dense in $W^{1,p}(M)$ for all such $p$. This yields some duality results between $W^{1,p}(Ω,\partial Ω)$ and $W^{-1,p'}(Ω)$ in the case where $1< p\le p_Ω$ and $Ω$ is a bounded subanalytic open subset of $\mathbb{R}^n$, and consequently that $W^{1,p}(Ω,\partial Ω)$ is reflexive for such $p$. As a byproduct, we deduce uniqueness of the (weak) solution of the Dirichlet problem associated with the Laplace equation. We then prove a version of Sobolev's Embedding Theorem for subanalytic bounded manifolds, show Gagliardo-Nirenberg's inequality (for all $p\in [1,\infty)$), and derive some versions of Poincaré-Friedrichs' inequality. We finish with a generalization of Morrey's Embedding Theorem.

math.AP↗

arXiv2024

A remark on $\mathscr{C}^\infty$ definable equivalence

We establish that if a submanifold $M$ of $\mathbb{R}^n$ is definable in some o-minimal structure then any definable submanifold $N\subset \mathbb{R}^n$ which is $\mathscr{C}^\infty$ diffeomorphic to $M$, with a diffeomorphism $h:N\to M$ that is sufficiently close to the identity, must be $\mathscr{C}^\infty$ definably diffeomorphic to $M$. The definable diffeomorphism between $N$ and $M$ is then provided by a tubular neighborhood of $M$.

math.AG↗

arXiv2024

Uniform Poincaré inequality in o-minimal structures

We first define the trace on a domain $Ω$ which is definable in an o-minimal structure. We then show that every function $u\in W^{1,p}(Ω)$ vanishing on the boundary in the trace sense satisfies Poincaré inequality. We finally show, given a definable family of domains $(Ω_t)_{t\in \mathbb{R}^k}$, that the constant of this inequality remains bounded, if so does the volume of $Ω_t$.

math.AP↗

arXiv2022

Regular vectors and bi-Lipschitz trivial stratifications in o-minimal structures

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which is locally definably bi-Lipschitz trivial along the strata. This result is obtained as a byproduct of two foregoing results of the author. The first one asserts that, given a family definable in an o-minimal structure, there is a regular vector, up to a definable family of bi-Lipschitz homeomorphisms. The second one is a bi-Lipschitz version of the famous Hardt's theorem. We give proofs of these two theorems that avoid the use of the real spectrum. The article recalls the basic facts and the results about Lipschitz geometry that are needed to understand the proofs, providing references.

math.LO↗

arXiv2022

On the Laplace equation on bounded subanalytic manifolds

We prove a trace formula for integration by parts on subanalytic bounded submanifolds of $\mathbb{R}^n$, possibly non closed. We also establish density results for $\mathbf{W}^{1,p}_\nabla (M)$, $M$ bounded subanalytic manifold, which is the space of the $L^p$ tangent vector fields $v$ on $M$ for which $\nabla v$ is $L^p$, where $\nabla$ is the divergence operator. We derive from these results some theorems of existence and uniqueness of solutions of the Laplace equation with Dirichlet and Neumann boundary type conditions. We then study the $p$-Laplace equation, for $p\in [1,\infty)$ large.

math.AP↗

arXiv2021

Trace operators on bounded subanalytic manifolds

We prove that if $M\subset \mathbb{R}^n$ is a bounded subanalytic submanifold of $\mathbb{R}^n$ such that $B(x_0,ε)\cap M$ is connected for every $x_0\in\overline{M}$ and $ε>0$ small, then, for $p\in [1,\infty)$ sufficiently large, the space $C^\infty(\overline{M})$ is dense in the Sobolev space $W^{1,p}(M)$. We also show that for $p$ large, if $A\subset \overline{M}\setminus M$ is subanalytic then the restriction mapping $ C^\infty(\overline{M})\ni u\mapsto u_{|A}\in L^p(A)$ is continuous (if $A$ is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of $\overline{M}$ is dropped.

math.FA↗

arXiv2021

Poincaré inequality on subanalytic sets

Let $Ω$ be a subanalytic bounded open subset of $\mathbb{R}^n$, with possibly singular boundary. We show that given $p\in [1,\infty)$, there is a constant $C$ such that for any $u\in W^{1,p}(Ω)$ we have $||u-u_Ω||_{L^p} \le C||\nabla u||_{L^p},$ where we have set $u_Ω:=\frac{1}{|Ω|}\int_Ω u.$

math.AP↗

arXiv2020

Multiple scaling limits of $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ multi-matrix models

We study the double- and triple-scaling limits of a complex multi-matrix model, with $\mathrm{U}(N)^2\times \mathrm{O}(D)$ symmetry. The double-scaling limit amounts to taking simultaneously the large-$N$ (matrix size) and large-$D$ (number of matrices) limits while keeping the ratio $N/\sqrt{D}=M$ fixed. The triple-scaling limit consists in taking the large-$M$ limit while tuning the coupling constant $λ$ to its critical value $λ_c$ and keeping fixed the product $M(λ_c-λ)^α$, for some value of $α$ that depends on the particular combinatorial restrictions imposed on the model. Our first main result is the complete recursive characterization of the Feynman graphs of arbitrary genus which survive in the double-scaling limit. Next, we classify all the dominant graphs in the triple-scaling limit, which we find to have a plane binary tree structure with decorations. Their critical behavior belongs to the universality class of branched polymers. Lastly, we classify all the dominant graphs in the triple-scaling limit under the restriction to three-edge connected (or two-particle irreducible) graphs. Their critical behavior falls in the universality class of Liouville quantum gravity (or, in other words, the Brownian sphere).

math-ph↗

arXiv2020

On the large $D$ expansion of Hermitian multi-matrix models

We investigate the existence and properties of a double asymptotic expansion in $1/N^{2}$ and $1/\sqrt{D}$ in $\mathrm{U}(N)\times\mathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $N\times N$ matrices transform in the vector representation of $\mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $η(h)$ on the maximum power $D^{1+η(h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ in the large $N$ expansion. We conjecture that $η(h)=h$ in a large class of models. In the case of traceless Hermitian matrices with the quartic tetrahedral interaction, we are able to prove that $η(h)\leq 2h$; the sharper bound $η(h)=h$ is proven for a complex bipartite version of the model, with no need to impose a tracelessness condition. We also prove that $η(h)=h$ for the Hermitian model with the sextic wheel interaction, again with no need to impose a tracelessness condition.

hep-th↗

arXiv2020

Melonic Turbulence

We propose a new application of random tensor theory to studies of non-linear random flows in many variables. Our focus is on non-linear resonant systems which often emerge as weakly non-linear approximations to problems whose linearized perturbations possess highly resonant spectra of frequencies (non-linear Schrödinger equations for Bose-Einstein condensates in harmonic traps, dynamics in Anti-de Sitter spacetimes, etc). We perform Gaussian averaging both for the tensor coupling between modes and for the initial conditions. In the limit when the initial configuration has many modes excited, we prove that there is a leading regime of perturbation theory governed by the melonic graphs of random tensor theory. Restricting the flow equation to the corresponding melonic approximation, we show that at least during a finite time interval, the initial excitation spreads over more modes, as expected in a turbulent cascade. We call this phenomenon melonic turbulence.

math-ph↗

arXiv2020

Poincaré duality for $L^p$ cohomology on subanalytic singular spaces

We investigate the problem of Poincaré duality for $L^p$ differential forms on bounded subanalytic submanifolds of $\mathbb{R}^n$ (not necessarily compact). We show that, when $p$ is sufficiently close to $1$ then the $L^p$ cohomology of such a submanifold is isomorphic to its singular homology. In the case where $p$ is large, we show that $L^p$ cohomology is dual to intersection homology. As a consequence, we can deduce that the $L^p$ cohomology is Poincaré dual to $L^q$ cohomology, if $p$ and $q$ are Hölder conjugate to each other and $p$ is sufficiently large.

math.AG↗

arXiv2019

New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications

Large $N$ matrix models play an important role in modern theoretical physics, ranging from quantum chromodynamics to string theory and holography. However, they remain a difficult technical challenge because in most cases it is not known how to perform the sum over planar graphs, which dominate the models at large $N$. In this thesis, we study large $D$ matrix models, which provide a framework to build new limits for matrix models in which the sum over planar graphs simplifies when $D$ is large. The basic degrees of freedom are real matrices of size $N\times N$ with $r$ additional indices of range $D$. They can be interpreted as a real tensor of rank $R=r+2$ with indices of different ranges, making a compelling connection with tensor models. We define a new large $D$ limit for the sum over Feynman graphs of fixed genus in matrix models, based on an enhanced large $D$ scaling of the coupling constants. Then, we show that the resulting large $D$ expansion is well-defined and organized according to a half-integer called the index. When $N=D$, the result provides a new large $N$ limit for general $\text{O}(N)^R$ invariant tensor models. In the large $D$ limit, the sum over planar graphs of large $N$ matrix models simplifies to a non-trivial sum over generalized melonic graphs. This class of graphs extends the one obtained in tensor models with standard scaling and allows for a wider class of interactions, including all the maximally single-trace terms. The general classification of generalized melonic graphs remains an open problem. However, in the case of the complete interaction of order $R+1$ for $R$ a prime number, we identify them in detail and demonstrate that they exhibit the same important features as the SYK model with $q=(R+1)$-fold random interactions, including the emergent conformal symmetry in the infrared regime and maximal chaos.

hep-th↗