Papers

Let $\Gamma$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(\Gamma)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\infty$ and $f$ belonging to the noncommutative $L^p$ space $ L^p(\hat \Gamma)$, we obtain convergence in the $L^p$ norm. The proof combines uniformly bounded finite-support Fourier cutoffs with an elementary Bernoulli even-moment estimate. We give a self-contained free-group argument, then establish a general criterion and verify it for the two classes above. The operator-norm conclusion establishes the Révész property for these groups. In particular, it confirms the polynomial-growth conjecture of Hamm, Hayes, and Petrosyan and disproves their conjecture that the free group $\F$ does not have this property.

We present a new perspective on integral operator kernels given by a nonlinear function of the distance. This perspective is useful since it permits one to quasilinearize the nonlinear transformations acting on the tensor Haar expansion of the distance, d(x,y). This leads to a new representation comprised of a principal and residual term with desirable features; namely, enhanced local regularity of the principal component (and in certain situations rapid off-diagonal decay leading to O(N) storage) and a residual expansion of improved approximation as a consequence of enhanced global regularity. Numerical experiments on the potential kernel are included

We study three classical problems in the theory of linear second-order uniformly elliptic equations: (i) gradient estimates for the Dirichlet problem, (ii) estimates and asymptotics for the first eigenvalue of an elliptic operator, and (iii) the Landis conjecture on exponential decay. Our main results give versions of (i) in which the constant is sharply specified in terms of the norms of the coefficients of the operator and the size of the domain; and use these to strongly improve on known results for (ii) and (iii) by allowing both more general operators and weaker regularity assumptions on the coefficients, and by giving quantitative estimates. The proofs use a unified approach, relying on three ingredients: interior and boundary Harnack inequalities with sharp constants, duality arguments, and a $C^1$ estimate based on a rescaling procedure. The sharpness of the gradient and spectral estimates is demonstrated through various counterexamples.

Let $X$ be a normed space and $S_X$ be the unit sphere of $X$. We say that $X$ has Property (H) if there exist two increasing sequences $V_j\subset X$ and $H_j\subset \ell_2$, $j=1,2,\ldots$, of finite dimensional subspaces such that $V=\bigcup_j V_j$ is dense in $X$, and there is a uniformly continuous mapping $\psi:S_V\to S_{\ell_2}$ such that the restrictions $\psi|_{S_{V_j}}$, $j=1,2,\ldots$, are isomorphisms from $S_{V_j}$ onto $S_{H_j}$. The Kasparov-Yu problem asks whether the Banach space $c_0$ of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that $c_0$ admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum $\mu_n$ of the Lipschitz constants of all nonzero-degree maps from $S_{\ell_\infty^n}$ to $S_{\ell_2^n}$ satisfies $\lim_{n\to\infty} \mu_n/\log n=1/2$. (2) Let $\lambda_n$ and $C_n$ denote the infima of the Lipschitz constants of homeomorphisms from $S_{\ell_\infty^n}$ onto $S_{\ell_2^n}$ and from $B_{\ell_\infty^n}$ onto $B_{\ell_2^n}$, respectively. Then $\lim_{n\to\infty} \lambda_n/\log n=\lim_{n\to\infty} C_n/\log n=1/2$. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.

We identify the sharp iterated-logarithmic thresholds at the critical exponent for quantitative measure equivalence and orbit equivalence between integer lattices. To be more precise, let $n>m$ be two positive integers, $\alpha$ be a positive number and $( \beta_j)_{j \geqslant 1}$ be a finitely supported sequence of non-negative numbers. We show that there is a quantitatively $t^{\alpha}\cdot \prod_{j \geqslant 1} ( \log^{(j)}{t} )^{-\beta_j} $-integrable measure equivalence from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ if and only if either $\alpha<m/n$ or $\alpha=m/n$ and there is $j_* \geqslant 1$ such that $\beta_j= 1$ for all $1 \leqslant j < j_*$ and $\beta_{j_*} > 1$. Here $\log^{(j)}$ is the $j$-fold iterated logarithm. The same characterization holds for quantitative orbit equivalence. This characterization greatly strengthens the previous best-known result, due to the work of Delabie, Koivisto, Le Maître and Tessera (2022) and the work of Correia (2025), which asserts that there is a $t^{\alpha}$-integrable ($\alpha>0$) measure equivalence (or orbit equivalence) from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ if and only if $\alpha<m/n$. In particular, our result solves, in much stronger forms, two open problems posed respectively by Delabie, Koivisto, Le Maître and Tessera, and by Naryshkin and Petrakos.

We construct metrics with $k$-positive Ricci curvature on $\mathbb{T}^k\times \mathbb{S}^m$, for $k,m\geq2$, arbitrarily close to the standard flat-round product in the smooth topology. The two-torus case disproves Wolfson's conjecture that every closed manifold with two-positive Ricci curvature has virtually free fundamental group. We also establish a local obstruction to two-positive Ricci curvature near closed backgrounds carrying three orthonormal parallel one-forms.

We establish the worst-case hardness of several ideal lattice problems (including SVP and CVP) in the $\ell_2$ norm by providing a dimension-preserving, deterministic polynomial time reduction from their generic lattice versions. The reduction constructs an ideal lattice in the canonical embedding of a number field that approximates some input lattice up to scaling and orthogonal transformation. The integers defining the ideal and the ambient number ring, in particular its discriminant, are all polynomial in bit length relative to the generic input lattice. Furthermore, the ideal is invertible, the ring is monogenic, and the number field is totally real. If the number ring is also required to be a full ring of integers, the reduction conjecturally succeeds in bounded-error quantum polynomial time.

We solve the robust ten martini problem: Cantor spectrum with no arithmetic restriction on the irrational frequency for Type I analytic one-frequency quasiperiodic Schrodinger operators. We introduce Type I through a new acceleration-type quantity with stability under analytic perturbations. This yields a global open class of analytic potentials whose associated operators have Cantor spectrum for every irrational frequency.

Statistical precision and scientific relevance operate on different scales. In regular problems, sampling uncertainty contracts at rate $n^{-1/2}$, whereas the magnitude below which an effect is scientifically negligible may be fixed or may vary with information. Let $\Delta_n$ denote a relevance threshold and $I_0$ Fisher information, and define $\lambda_n=\sqrt{nI_0}\Delta_n$. For separated negligible and meaningful parameter classes, we establish the minimax lower bound $\liminf R_n^*\ge 2\{1-\Phi(\varepsilon\kappa)\}$ when $\lambda_n\to\kappa$. If $\lambda_n\to0$, the experiments merge and consistent classification is impossible. If $\lambda_n\to\kappa\in(0,\infty)$, the problem converges to a Gaussian-shift decision problem. We characterize the exact minimax rule and risk in that limiting composite problem. As $\kappa\downarrow0$, the optimal cutoff converges to one statistical-error unit and the optimal improvement over trivial risk is $O(\kappa^2)$, while a simple relevance-boundary rule improves only at $O(\kappa^3)$; as $\kappa$ grows, that rule becomes asymptotically minimax. If $\lambda_n\to\infty$, consistent classification is attainable under a uniform estimation-resolution condition, verified for Gaussian and Bernoulli models. For $\Delta_n=dn^{-\gamma}$, the critical rate is $\gamma=1/2$. We also show that moving asymmetric relevance regions depend on standardized distances to their two boundaries rather than total width alone. For heterogeneous true effects, we derive a significance-saturation limit and show that point-null significance can be most selective for scientifically relevant effects at intermediate information. The framework characterizes when scientific relevance is statistically resolvable through a common information scale.

Let $0<s<1$ and $A$ be a constant symmetric positive definite matrix. Given the quadratic form $$ Q_A(u)=\int_{\R^N} A\nabla^s u(x)\cdot \nabla^s u(x)\,dx, $$ where $\nabla^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet form. In strike contrast with the local case $s=1$, positive definiteness of a constant matrix is not sufficient for the Markov property when $0<s<1$. We also describe the associated symmetric Markov semigroup and its $L^p$ realizations, prove the existence of a smooth nonnegative heat kernel and establish a bounded $H^\infty$ functional calculus.

Proximal bundle methods (PBMs) are classical algorithms for nonsmooth convex optimization. Existing analyses of the classical PBM couple the null steps with the descent test. This coupling obscures how the bundle updates approximate the proximal subproblem. In this work, we consider composite objectives $F=f+h$ and view each null-step cycle as an inner bundle subroutine, called $\mathtt{ProxBundle}$. We analyze $\mathtt{ProxBundle}$ independently of any stopping criterion under general bundle model conditions and show that it automatically adapts to Hölder smoothness. Combining this inner-loop analysis with the descent-step analysis yields sharper complexity bounds for the classical PBM. For any fixed proximal parameter, its overall complexity is $\mathcal O\big(\epsilon^{-\frac{3-\nu}{1+\nu}}\big)$ for $\nu\in[0,1)$ and $\mathcal O(\epsilon^{-1})$ for $\nu=1$, where $\nu$ is the Hölder smoothness exponent. Choosing the proximal parameter proportional to $\epsilon$ improves the rate to $\mathcal O\big(\epsilon^{-\frac{2}{1+\nu}}\big)$ for $\nu \in [0,1)$. These are the first guarantees under Hölder smoothness with $\nu\in(0,1)$ for the classical descent test. We further introduce an absolute model-error test. The resulting PBM variant admits a clean inexact proximal-point analysis, and for any fixed proximal parameter, achieves the same complexity $\mathcal O\big(\epsilon^{-\frac{2}{1+\nu}}\big)$. Overall, our analysis separates the roles of the descent and null steps and gives a modular understanding of PBMs across different~tests.

We say that a function f can be represented by a Shannon integral if it can be written as the integral of sin(xs)/(xs), with respect to some finite measure. We develop a theory of Shannon integral representations for positive definite functions and show that under mild conditions f can be written as sum of two positive definite functions that constitute solutions to an additive Reimann_Hilbert problem on real line.

We introduce loss-conditioned state execution, a model-agnostic method that decides whether to execute a world model's fixed feasible proposal or retain the current state. Predictive informativeness alone, however, does not establish whether an update will reduce downstream loss. Occurrence ranking can approach perfection while persistence remains the unique absolute-loss Bayes action. Two transition laws can also share occurrence information and conditional variance yet require opposite absolute-loss decisions. We formalize state movability as the existence of a loss-reducing feasible correction and distinguish it from the benefit of a particular proposal. Our method constructs a loss-specific feasible proposal from a predictive distribution and evaluates its groupwise bounded-loss gain over persistence on independent calibration units. The proposal is executed only in groups with a positive simultaneous lower confidence bound. For fixed proposals and groups with bounded unit losses, we prove that every accepted group has lower expected loss than persistence with high probability when calibration units are i.i.d. draws from the target population. Experiments on public forecasting and action-conditioned dynamics benchmarks show supported updates and a trade-off between certification and coverage. On 28,684 held-out M4 Monthly series, the method executes the proposal for 14.0% of series and achieves bounded loss 0.588, compared with 0.599 for persistence and 0.621 for always executing the proposal. The paired 95% bootstrap intervals for both comparisons lie below zero. In constrained forecasting of six unhealthy-inventory types from JD$\mbox{.}$com, a leading e-retailer in China, strong occurrence-ranking signal coexists with a loss-based preference for persistence, illustrating why event predictability and state execution must be evaluated separately.

We study propagation of chaos for decoupled mean-field forward-backward stochastic differential equations whose generators depend on the empirical laws of the forward states, backward values and diagonal martingale integrands. Under monotonicity and Lipschitz assumptions, synchronous coupling gives quantitative estimates, including an $m$-particle squared Wasserstein bound of order $m/n$ for a system of $n$ particles interacting through finitely many statistics. In the Markovian setting, assuming a sufficiently regular classical decoupling field, we obtain two sharp refinements. For constant invertible diffusion and first-order cancellation of the field's measure dependence along the limiting law flow, the squared Wasserstein error is of order $m^2/n^2$, on continuous-path space for the values and on $L^2$ for the diagonal integrands. Without imposing this cancellation, smooth weak errors have order $n^{-1}$ for every fixed marginal, allowing variable and possibly degenerate diffusion. The weak estimate is uniform on a fixed time interval for the values and integrated in time for the integrands. The argument compares the interacting BSDE with an empirical evaluation of the decoupling field, retaining the full martingale representation and controlling feedback through both backward laws. It yields a joint-path Wasserstein transfer bound with intrinsic squared error $m/n^2$, off-diagonal integrand estimates, and a weak-error transfer principle with additive error $n^{-1}$. Explicit models with feedback through both backward laws verify the cancellation assumptions. Examples distinguish the intrinsic backward error from the forward law error and establish matching lower bounds for each backward component.

We study exact community recovery in the Degree-Corrected Poisson Stochastic Block Model (DC-SBM) through the maximum profile likelihood estimator. Our main contribution is identifying an explicit complexity constant $C(\pi,S)$, which closely captures the exact recovery threshold. This constant combines a weighted Chernoff–Hellinger separation between communities with a Kullback–Leibler correction induced by the degree-normalized block volumes. We rigorously prove that $C(\pi,S)>1$ ensures the strong consistency of the maximum profile likelihood estimator, yielding exact community recovery down to the logarithmic sparsity scale. We conjecture that $C(\pi,S)=1$ represents the sharp information-theoretic threshold, providing an upper bound on the exact-recovery transition, and we discuss analogies with the binary SBM and binary DC-SBM cases. Additionally, we establish weak consistency when the average degree diverges, offering a unified picture across sparse and dense regimes. The proofs require new control of the profile likelihood in the presence of heterogeneous degree parameters, whose node-specific weights fundamentally alter the geometry and concentration of the recovery problem.

Let $X$ be an oriented topological $4$-manifold with boundary. We establish a Dax isomorphism for the fundamental group of the space of embedded arcs in $X$. Here, the embedding space can be either the simplicial set of locally flat embeddings or the space of topological embeddings endowed with the compact-open topology. When $X$ is smooth, we show that the space of smooth arcs and topological arcs have canonically isomorphic fundamental groups, and the topological Dax isomorphism agrees with the smooth Dax isomorphism. As a consequence, the homomorphisms that arise in the smooth Dax isomorphism do not depend on the smooth structure.

We develop a Gaussian comparison theory for posterior inference in non-Gaussian high-dimensional models. The framework allows for model misspecification and does not require the posterior distribution itself to be approximately Gaussian. Working directly with likelihood processes on separable function spaces, we establish explicit nonasymptotic free energy bounds controlled by first two moment discrepancies and local third moments. A perturbation-and-convexity argument transfers these comparisons to sandwich bounds for posterior mean squared error and posterior variance, including one-sided conclusions at nondifferentiable phase transitions. In high-dimensional product models, these conditions reduce to local moment controls and a canonical feature-radius scaling. Under weak regularity requirements, the resulting universality theory applies to a broad class of likelihood models, ranging from exponential families to more general regular local log-likelihoods. The framework is illustrated on sparse Bernoulli hypergraph inference.

In the compressible primitive equations, gravity is not merely a source term: through the hydrostatic balance it forces, it organizes the system's energy geometry. The hydrostatic balance forces the sum of the barotropic enthalpy and the geopotential to be independent of the vertical coordinate an intrinsic rst hydrostatic integral, coinciding with the classical Montgomery potential of geophysical uid dynamics and testing the continuity equation against it gives the total mechanical energy identity, with no separate estimate on the vertical velocity. The core of this note is an augmented hydrostatic identity, obtained by testing the momentum equation against a BreschDesjardins-type augmented velocity: dierentiating the rst integral again produces two structural functions governing its cross terms, whose vanishing cases are characterized exactly here one vanishes identically if and only if the pressure is quadratic, the other if and only if it is isothermal, no law annihilating both. The resulting augmented identity also isolates a coupling involving the horizontal gradient of the vertical velocity that is not controlled by the energetic structure developed here, a limitation of this direct route to closing a BreschDesjardinstype entropy, not an impossibility theorem. For the isothermal law, the vanishing of the second function forces the vertical density prole to be a pure exponential, exactly the factorization underlying the Ersoy NgomSy change of variables the multiplicative expression of the Montgomery rst integral at that one law on which basis we give a detailed return-to-original-variables argument for the existence theorem of Wang, Dou and Jiu.

We give a new and shorter proof of a quantitative inverse theorem for the Gowers $U^4$-norm in the setting of high-dimensional vector spaces over finite fields.

We prove the degree-seven case of the Hermite–Joubert problem in characteristic zero: if $F$ is a field of characteristic zero and $E/F$ is a field extension of degree seven, then $E$ is generated by an element $a$ with $\text{tr}_{E/F}(a)=\text{tr}_{E/F}(a^{3})=0$, that is, with minimal polynomial of the form $\lambda^{7}+c_{2}\lambda^{5}+c_{4}\lambda^{3}+c_{5}\lambda^{2}+c_{6}\lambda+c_{7}$, where the $c_i$'s belong to $F$. This is given by an explicit formula: a covariant of the binary septic of coefficient degree seven and order five, evaluated at a generator $\theta$ and divided by the derivative of its minimal polynomial evaluated at $\theta$. We also announce the general theorem, which will be proved in a companion paper in preparation: over every infinite field, in every characteristic, every étale algebra of degree seven contains a primitive element $a$ with $c_{1}(a)=c_{3}(a)=0$. Moreover, every field extension of degree seven of an arbitrary field has a generator $a$ with $c_{1}(a)=c_{3}(a)=0$.

We introduce a poset of weighted hypergraphs on a fixed vertex set which has a remarkable property: the Möbius values of the intervals associated to any hypergraph give, up to sign, the coefficients of the $h$-polynomial of its nestohedron. We exploit this relation to provide two new recurrence relations to compute $h$-polynomials of nestohedra.

It is shown that an infinite dimensional, simple, unital, monotracial C*-algebra with strict comparison with respect to its trace is selfless.

In this paper, we resolve the convolution inequality problem posed by Naor and Schechtman in \cite[Question 6.1]{N-S2016} (see also \cite{Na2016}). More precisely, we establish the following strong convolution inequality on products of finite cyclic groups: For each $1\leq p<\infty$ there exists $c_{p}>0$ such that for every $n$, $m\in \mathbb{N}$ the following holds \[ \begin{split} &\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|E_{\{1,\cdots,n\}}f(x+\varepsilon)-E_{\{1,\cdots,n\}}f(x-\varepsilon)\right|^{p}\\ \leq& c^{p}_{p}\sum_{\varepsilon\in \{-1,1\}^{n}}\sum_{x\in \Z}\left|\varepsilon_{j}\left[E_{\{1,\cdots,n\}\setminus\{j\}}f(x+e_{j})-E_{\{1,\cdots,n\}\setminus\{j\}}f(x-e_{j})\right]\right|^{p}, \end{split} \] $E_{A}f(x)=\frac{1}{2^{n}}\sum_{\delta\in \{-1,1\}^{n}}f\left(x+\sum_{j\in A}\delta_{j}e_{j}\right)$ for every $A\subseteq \{1,\cdots,n\}$. The growth order we obtain here $c_{p}=O(p)$ as $p\to \infty$. For the endpoint $p=\infty$, we show by a counterexample that the strong convolution inequality fails for the $L_{\infty}$, and we prove that the appropriate endpoint for $p=\infty$ is the martingale $BMO$ space.

Nested cubature rules, in which each refinement retains all previously used nodes and thus reuses earlier function evaluations, are natural in multilevel and adaptive integration. For every fixed $s>d/2$, we show that the equal-weight QMC integration rate on $\mathbb S^d$ is compatible with such nested point sets. In the subcritical range $d/2<s<d$, cumulative unions of geometrically growing QMC blocks yield nested QMC design sequences whose successive cardinality ratios converge to any prescribed $\rho>1$. At and above the critical index $s=d$, where the block-averaging estimate no longer yields the optimal rate, we prove an equal-weight completion theorem based on low-frequency discrepancy cancellation. A block-sensitive estimate sharpens the iteration and yields nested QMC designs for all $s\ge d$ with $N_{j+1}\lesssim(j+1)^{2s/d-1}N_j$. Every prescribed finite point set also admits an optimal-rate completion.

Dissipativity theory enables the stability analysis of complex, interconnected systems through the physical properties of their components. Celebrated examples of such stability results include the small gain and passivity theorems. In this paper, we prove a unifying stability theorem for dissipative linear time-invariant (LTI) systems. We use the behavioral approach to express storage functions and supply rates in terms of quadratic differential forms (QDFs). Our main result establishes conditions for the stability of an interconnection of two LTI behaviors that are both dissipative with respect to general, dynamic supply rates. As special cases, we recover existing results for static supply rates, such as those capturing finite $L_2$-gain and passivity, as well as dynamic ones capturing, for example, delta dissipativity. The key idea of the paper is to construct a Lyapunov function as a sum of storage functions and a coupling QDF, thereby allowing for a richer class of Lyapunov functions than existing works dealing with just sums of storage functions.

Entanglement can reduce the communication required for coding tasks, but establishing the minimum achievable cost is essential to understanding its limits. We address this question in a zero-error source-coding task where Alice receives a word and Bob knows an unordered pair of candidates containing it. Alice does not know the pair and must enable Bob to identify her word without error using shared entanglement and one classical message. The candidates satisfy a balanced-difference promise: for words in $\mathbb{Z}_q^n$ with $n=q\ell$, each residue modulo $q$ occurs exactly $\ell$ times in their coordinatewise difference. For all integers $q\geq2$ and $\ell\geq1$, we prove that the task requires exactly $n$ messages when $(q-1)\ell$ is even and two messages when it is odd. These minima allow arbitrary finite-dimensional shared states independent of the inputs and arbitrary local measurements. In even parity, this establishes optimality of an existing entanglement-assisted protocol. In odd parity, an explicit deterministic protocol achieves the optimum of one bit without entanglement. Our proof combines Fourier analysis with a combinatorial counting argument to determine the smallest eigenvalue of the associated graphs. In even parity, this resolves the spectral assertion of Cao et al.'s Conjecture 6.3 for balanced cyclic generalized Hadamard graphs. Together with an explicit odd-parity bipartition, this determines the quantum chromatic number as $n$ in even parity and $2$ in odd parity, where the classical chromatic number is also $2$. All lemmas, theorems, and corollaries are formalized and verified in Lean.

We construct a five-parameter quadratic algebraic generating function whose coefficient sequence has prescribed initial Hankel determinants $(\lambda,m,r,\eta)$ and whose Hankel transform belongs to the Somos-4 family $A(1,\tau)$. The construction starts from a Stieltjes continued fraction whose coefficients are generated by an alternating recurrence compatible with the Somos-4 relation. We derive an explicit quadratic equation for the resulting generating function and give the corresponding coefficient recurrence. A subtle feature of the construction is the nonuniqueness of a coefficient sequence determined solely by its ordinary Hankel transform. To select a canonical representative, we additionally prescribe the shifted Hankel determinants obtained from the same Somos-4 orbit advanced by two indices. This companion condition determines the odd and even Stieltjes coefficients separately and removes the remaining freedom in the continued-fraction representation. We also analyze the two algebraic branches at the origin, where they coalesce, and derive a desingularized recurrence for the coefficients. The resulting construction provides a direct algebraic generating-function realization of a general five-parameter family of Somos-4 Hankel transforms.

We develop notions of covariance and principal component analysis (PCA) for probability measures on Riemannian manifolds of bounded geometry and a discrete counterpart for distributions on the vertex sets of weighted simple graphs. Rather than anchoring variation at the Fréchet mean, whose uniqueness and usefulness can fail in this context, we consider variation about every point. This leads to a representation of each point on the manifold by a vector field derived from the heat kernel and thus to a map of the underlying manifold into a Hilbert space of vector fields, in which covariance and PCA carry over from the Euclidean setting. A distinguishing feature of this formulation is that one typically obtains infinitely many principal components, capable of capturing highly nonlinear geometric features and patterns of variation. We prove an embedding theorem for the mapping into the space of vector fields and develop two computational reductions of the theory: RiePCA, a finite-dimensional reduction based on vector fields supported on finitely many points, and GraphPCA, a discrete formulation for measures on weighted graphs in which vector fields assign orientations and magnitudes to edges. Numerical examples and experiments illustrate the behavior of the proposed methods on both synthetic and real data.

Let $\Gamma,\Delta$ be finite simplicial graphs, and assume that $\Gamma$ has no induced squares. We prove that the right-angled Artin group $A(\Delta)$ measurably embeds into $A(\Gamma)$ if and only if $A(\Delta)$ embeds as a subgroup in a graph product of free abelian groups over $\Gamma$. This in turn has a graph-theoretical characterisation which can be checked algorithmically. If additionally $A(\Gamma)$ and $A(\Delta)$ have cohomological dimension equal to two and $A(\Gamma)$ is not isomorphic to $\mathbb{Z}\times F_n$, we get that $A(\Delta)$ measurably embeds into $A(\Gamma)$ if and only if it embeds as a subgroup in $A(\Gamma)$. Our proof relies on the following algebraic statement. Let $K_1,\dots,K_n$ be a family of pairwise disjoint cliques in the square-free graph $\Gamma$, and let $g_1,\dots,g_n$ be elements of $A(\Gamma)$ with respective parabolic supports $A(K_1),\dots,A(K_n)$. Then the subgroup of $A(\Gamma)$ generated by $g_1,\dots,g_n$ is a right-angled Artin group, where the only relations impose that $g_i$ and $g_j$ commute if $K_i\cup K_j$ is contained in a clique of $\Gamma$. The same holds true more generally for pairwise disjoint cliques $K_1,\dots,K_n$ in the extension graph $\Gamma^{\mathrm{ext}}$, up to possibly replacing the elements $g_i$ by powers.

We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi > 1$, we prove almost-sure convergence of $\lambda_n (\mathbf{R}_n)$ to the lower Marčenko–Pastur edge $\lambda_- = (1 - \sqrt{\phi})^2$ without additional moment assumptions. In the wide regime $\phi < 1$, we establish a phase transition at the third-order tail scale. If $t^3 \mathbb{P}\{\lvert \xi \rvert > t\} \to 0$ as $t \to \infty$, the smallest eigenvalue $\lambda_{p_n} (\mathbf{R}_n)$ converges in probability to $\lambda_-$. While if $t^3 \mathbb{P}\{\lvert \xi \rvert > t\} \to \infty$, then $\lambda_{p_n} (\mathbf{R}_n)$ converges in probability to zero. At the critical scale $t^3 \mathbb{P}\{\lvert \xi \rvert > t\} \to \kappa \in (0, \infty)$, the point process of eigenvalues in the lower gap $(0, \lambda_-)$ converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of $\lambda_{p_n} (\mathbf{R}_n)$, which has a continuous density on $(0, \lambda_-)$ and a positive atom at $\lambda_-$.