Papers

In this article we investigate the (non)-coincidence of critical parameters for various related percolation problems. More precisely, for the random walk loop soup we show that on $\mathbb Z^d$, $d\ge 5$, the critical parameters for the percolation problems differ on the discrete graph and the respective metric graph. Moreover, on trees we deduce an analogous statement as well as the coincidence of the critical parameters for percolation and susceptibility for a more general class of percolation problems, the so-called Poisson zoo. Along the way we develop the useful notion of sensitivity to Bernoulli enhancements of such percolation problems with long range correlations, which builds on previously developed enhancement ideas.

We introduce the notion of "$k$-thresholds” and show that Talagrand's discrete convexity conjecture is equivalent to the assertion that, for some universal integer $k \ge 2$, the $k$-threshold of every increasing family is at most a universal constant times its expectation threshold. We prove a reduction theorem that bounds the $k$-threshold of any increasing graph property in terms of ordinary thresholds of graphs in suitable decompositions of its members. As a consequence, we determine, up to a constant factor, the $k$-threshold of every fixed graph in terms of a natural $k$-density parameter. We also prove that $k=2$ suffices for several classical spanning graph containment properties. More generally, we establish the conjectured comparison between $k$-thresholds and expectation thresholds for broad classes of graph containment properties whose target graphs have low degeneracy.

We study the near-origin behavior on $[0,T]$ of the singular diffusion whose transition density is given by a Doob transform of the integral kernel of the semigroup generated by the $d$-dimensional Schrödinger operator $L^{\gamma}$ with a one-point potential at the origin, where the driving family is the ground state of $L^{\gamma}$ and $d\in\{2,3\}$. We construct a submartingale whose increasing component grows only at times when the diffusion visits the origin. Using this submartingale, we show that the diffusion hits the origin with positive probability and that, conditionally on hitting the origin by time $T$, the first hitting time has a truncated generalized inverse Gaussian (GIG) distribution. We further study the dynamics under conditioning to avoid the origin: under the conditional law, the diffusion is not a standard Brownian motion, but instead admits a representation in terms of a regularized drift and a continuous martingale. While these properties are known in dimension two, the present submartingale-based approach provides an alternative verification and treats dimensions two and three in a unified manner.

The Renormalization Group (RG) is a powerful and versatile framework for analyzing complex physical systems. Here, we reinterpret it as a stochastic process that explores physical phase spaces, scale by scale, progressively revealing finer details of small-scale structures. This perspective establishes a natural connection to other random exploration processes, such as the Schramm-Loewner evolution, and is more suited for a probabilist audience. It links RG concepts such as RG transformations, effective actions, etc, to usual probabilistic tools such as conditional expectation values, martingales, etc, but also makes contact with stochastic quantization.

This paper establishes sharp dimension-free Frobenius-norm concentration inequalities for sample moment tensors. Our bounds are optimal over the sub-Gaussian class, while for Gaussian data we obtain matching two-sided estimates. We also identify a parity effect: the intermediate Gaussian and sub-Gaussian scales coincide at odd tensor orders but differ at even orders. Our second main result, which supplies the weak-moment estimate behind these bounds, proves a dimension-free moment bound for Hilbert-valued polynomials under Gaussian convex domination. The proof of this bound combines a recently developed structured martingale coupling theorem with new estimates for moment tensors of uniformly log-concave distributions.

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$-choices rule, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $\gamma_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $\gamma_0$ and $\gamma_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $\gamma_0$ and $\gamma_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $\gamma_0$ and $\gamma_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

This paper develops a path-space theory of entropy production for a class of stochastic evolution equations on infinite-dimensional Hilbert spaces. Since such spaces have no canonical Lebesgue reference measure, the usual finite-dimensional density formulas do not extend directly. We instead work relative to the invariant Gaussian measure of a reversible Ornstein–Uhlenbeck reference process. Combining an infinite-dimensional Girsanov transform, time reversal of the reference process, and the stationary Fokker–Planck equation relative to the Gaussian measure, we derive an explicit entropy-production formula in terms of an irreversibility field. On the natural test class, this field represents the difference between the forward and reversed nonlinear drifts. Under the standing assumptions, vanishing entropy production is equivalent to vanishing stationary probability current, self-adjointness of the generator in the invariant Hilbert space, detailed balance, and invariance of the stationary path law under time reversal. The reversible case is therefore characterized by a Gaussian-reference gradient structure for the nonlinear drift.

We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Latała. Our approach is based on information-theoretic ideas: lower bounds on the supremum of a Bernoulli process are translated to the fundamental limits of Bayesian estimation in a Cauchy additive channel. This leads to a new information-theoretic functional that characterizes Bernoulli-process suprema and plays a role analogous to Fernique's majorizing-measure functional for Gaussian processes. The same viewpoint yields a distributional strengthening: for any prescribed law of the index, we characterize the largest expected value attainable over all couplings of that index with the Bernoulli process. This extends to Bernoulli processes a phenomenon previously understood for Gaussian processes through the work of Fernique and Talagrand.

Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like $k$-means. Such methods are strongly connected to quantization problems, which aim to approximate general probability measures with discrete ones. For example, $k$-means corresponds to quantization with respect to the Wasserstein distance. While Wasserstein quantization clusters points within a fixed space, this paper studies Gromov-Wasserstein (GW) quantization, which additionally aims at clustering the ambient geometry of the space. We show existence of solutions to the GW quantization problem and give a characterization that justifies an analogue to the $k$-means algorithm (Lloyd's algorithm) to approximate them numerically. We further calculate the quantization rate for usual Euclidean geometries that are used in the GW context, and relate it to standard Wasserstein quantization rates. Finally, numerical experiments show that GW quantization opens up many modeling possibilities beyond normal clustering methods (e.g., for geodesic distances of 3D shapes or structured pruning of neural networks) and that the introduced algorithm leads to useful numerical solutions with approximation quality often in line with theoretically optimal rates.

We prove sharpness of the avalanche phase transition in the one-dimensional Bak–Sneppen model, for both the one-sided and isotropic update rules. In the one-sided model, the avalanche range is described by a nonlinear self-interacting jump process whose tails satisfy a triangular system. An extremal inequality for decreasing probability masses yields a polynomial lower bound on the range tail at the susceptibility threshold; this bound forces the appearance of infinite avalanches immediately above criticality and hence identifies the susceptibility and infinite-avalanche thresholds. A comparison of avalanche endpoints transfers this equality to the isotropic model. In both models, no infinite avalanche occurs at criticality, while the range has exponential moments at every strictly subcritical level. Combined with the range–duration and stationary-limit results of Meester and Znamenski, this identifies all three avalanche critical thresholds and proves the conjectured stationary product limit in both models.

We prove multiple central limit theorems for serial tail dependence estimators in heavy-tailed long memory linear time series. The main theoretical tools are two novel multivariate reduction principles for partial sums of heavy-tailed long memory linear time series, subordinated over sliding windows and above a threshold growing with sample size. This requires addressing several substantial difficulties, including handling a nonlinear, sample-size dependent, and multivariate subordination mechanism, the dependence between several overlapping linear processes, and the lack of higher-order moments of the marginal distribution. Despite these obstacles, our assumptions are mild and, in particular, the innovation process is allowed to have infinite variance. A key feature of our theory is that our second reduction principle holds uniformly in the threshold, allowing central limit theory for empirical extremograms with sample quantiles as thresholds. This question has received little attention in the literature on serial extremal dependence estimation even though the version of empirical extremograms with random thresholds is ubiquitous in practice. We compare our results in several respects with those that may be obtained under short-range dependence, thereby discovering markedly different convergence rates and limit laws in our long memory setting.

We study a Restart POMDP (Partially Observable Markov Decision Process) on a general Borel state space, where the controller either lets the hidden state evolve unobserved or restarts the system and observes the new state. Exploiting a sufficient-statistic representation consisting of the last observed state and the elapsed time since restart, we reduce the problem to a fully observed MDP. Under a natural one-step cost deterioration condition, we prove that optimal policies have a threshold structure in the elapsed time for both the discounted and total undiscounted cost criteria. When the state space is partially ordered and the kernel is stochastically monotone, we further show that the optimal threshold is nonincreasing in the state. For the average cost criterion, under additional assumptions of geometric ergodicity and domination of the transient gain, we establish analogous threshold results via the vanishing discount approach, after showing the uniform boundedness of the optimal thresholds and relative value functions.

Every statistic on laws with finite pth moment that is additive across independent risks and monotone in mean-preserving spreads depends only on an additive function of the mean when p is strictly less than 2, and only on such a function and a nonnegative multiple of the variance when p is no less than 2.

Underdamped Langevin diffusions model kinetic sampling and thermally driven inertial dynamics, but quantitative convergence is difficult when degenerate noise is coupled with a nonconvex superlinear force. We prove unit-prefactor exponential contraction in an explicit weighted Kantorovich cost under radial confinement, a hypocoercive Lyapunov condition, and \(|\nabla U(x)-\nabla U(y)|\leq L_1(1+|x|^\ell+|y|^\ell)|x-y|\), \(0\leq\ell\leq2\). No smallness condition is imposed on \(L_1\). At the critical exponent \(\ell=2\), a quartic lower bound on \(U\) is used together with an anisotropic position–velocity localization to obtain a nonempty endpoint regime. The proof combines regularized reflection–synchronous coupling, an exponential Lyapunov weight, tightness, and an occupation-time argument at the singular coupling direction. The result yields exponential convergence and uniqueness of the invariant Gibbs measure within the finite-cost class. It covers power-law confining potentials below quartic growth and nonconvex quartic Duffing potentials with cubic force.

Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $\phi$ and every $q\geqslant 1$, $$ \|\phi(G)\|_q\leqslant C\sqrt{\ln(en)+q}\,\|\phi(X)\|_q,\qquad \|\phi(X)\|_q\leqslant C\left(\sqrt{\ln(en)}+\psi(X)\sqrt q\right)\|\phi(G)\|_q. $$ Applied to support functions, this gives the sharp worst case order $C\sqrt{\ln(en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. Combining, at each level of Latala's dyadic chain, the strongest of the three quantitative $L_p$-Sudakov estimates used here yields $$ \left(\mathbb{E}\|Y\|^p\right)^{1/p}\leqslant C\left(n^{1/4}\sqrt{\ln(en)}\,\ln(e+\ln(en))\,\mathbb{E}\|X\|+\sigma_p(Y)\right) $$ whenever the weak moments of $Y$ are dominated by those of $X$. In a second direction, we study the generalized dual Sudakov problem for the self generated metrics associated with $Z_r(X)^\circ$ and prove dimension free packing estimates for $Z_p(X)$. We also obtain mean norm estimates for centroid bodies, factorization through arbitrary symmetric convex bodies and an affine dimensional refinement.

Let $M_n$ be the minimum spanning tree of the complete graph $K_n$ with i.i.d.\ uniform edge weights. For a fixed forest $F$ with connected components $T_1, \ldots, T_d$, we show that there exists a function $\Psi$ on finite trees such that $$ n^{|E(F)|} \mathbb{P}_n(F \subseteq M_n) \longrightarrow \prod_{i=1}^d \Psi(T_i). $$ We give a recursive description of $\Psi$ and calculate it explicitly for several small trees. For the star $S_k$ and the path $P_k$, we prove that $\Psi(S_k) \sim \zeta(2)^k$ and $\Psi(P_k) \sim k^2/12$, respectively. We also show that the expected size of a ball of radius $r$ is asymptotic to $r^3/36$, and give exponential tail bounds.

We consider fluctuations of the spin current in the stochastic XNOR process, a kinetically constrained hopping model in one spatial dimension. We formulate three conjectures with explicit amplitudes for the long-time current distributions: a Gaussian limit on the $t^{1/4}$ scale for homogeneous initial states with nonzero magnetization, a half-normal limit on the $t^{1/4}$ scale for domain-wall states with opposite magnetizations, and an M-Wright limit on the $t^{1/8}$ scale for homogeneous initial states at zero magnetization. Earlier work identified the tracer mechanism and anticipated the scaling exponents and limiting shapes in the domain-wall and zero-magnetization settings. Starting from the microscopic XNOR dynamics, we predict the missing amplitudes for the continuous-time process and extend the picture to homogeneous biased initial data. Simulations provide numerical support for all three conjectures without fitted parameters. A separate mathematical companion paper presents an extensively AI-generated candidate proof. Generative-AI tools were used in the research and writing workflow.

We study the limiting law of local field in the mean field Ghatak-Sherrington model with spin values $\{0,\pm1,\ldots,\pm S\}$ in the high-temperature regime. Using the cavity method and the quantitative overlap and self-overlap estimates of Sheng-Wu~\cite{SW24}, we prove a central limit theorem for the non-centered cavity field. As a result, we then identify the local field limiting law as a random finite mixture of Gaussian distributions. The proof is based on moment method and gives a quantitative error bound.

We propose a novel framework for modeling binary opinions (0 or 1) of individuals connected through a weighted directed network, where edge weights quantify interpersonal influence. Unlike classical models that assume complete access to previously expressed opinions, our framework allows individuals to update their biases using structured memory sets that capture limited and delayed information exchange. To analyze these opinion differences, we introduce a mathematically tractable notion of relative bias between pairs of individuals. The relative biases evolve according to a linear update rule involving past expressed opinions specified by the memory sets. We define the belief of an individual as the probability of expressing opinion 1 and derive a time-delayed dynamical system governing the evolution of network beliefs. We establish its asymptotic behavior and characterize its properties. The framework is further extended to networks containing bots, which maintain fixed biases while influencing neighboring individuals. We quantify the effect of bots by comparing the fixed points of the dynamics in their presence and absence. Finally, simulations illustrate the influence of memory, network structure, and bot interactions on the resulting opinion dynamics.

This is an introduction to probability, written with a quantum idea in mind, namely that the semicircle law comes first. We first discuss discrete probability, notably with the binomial and hypergeometric laws, positive and negative, and the Poisson and compound Poisson laws. Then we get into the continuous case, with the basics of the theory explained, and with as starting examples the exponential, semicircle and beta distributions. Afterwards, we investigate the central limits and normal variables, both real and complex, and with a look at Rayleigh laws, and hyperspherical laws too. Finally, we discuss a number of more specialized distributions, and more specialized techniques too, and we end with an introduction to random matrices and freeness.

Let U be uniformly distributed on the unit sphere. We develop a self-contained forward and inverse theory for the random width w_K(U) and brightness b_K(U) of three-dimensional convex bodies. For every full-dimensional polytope, a global spherical co-area formula expresses the width density as a finite sum of angular apertures determined by the normal fan of its difference body; in particular, the density is piecewise real analytic with a finite geometrically determined critical set. This theory yields exact densities for the width of the regular tetrahedron, resolving a question of Finch, and for the regular truncated octahedron, together with the tetrahedral brightness law and the equivalent rhombic-dodecahedral width law. On the inverse side, second- and third-order polarized cosine-transform moments reconstruct finite labelled direction systems whenever the observed triangles span the cycle space of the correlation graph; signed-graph switching describes the unavoidable ambiguity. In contrast, equal three-dimensional intrinsic volumes do not determine either the width law or the brightness law, even for centrally symmetric bodies. Removing the spatial rank constraint gives a dimension-free identifiability theorem for centered multivariate folded-normal vectors: pairwise absolute moments and an anchored family of triple absolute moments, comprising |m - 1|^2 labelled observations for a complete correlation graph, determine the correlation matrix up to diagonal sign conjugacy without fourth-order moments. A harmonic decomposition further identifies the degree-two variance contribution as a constant multiple of the squared Frobenius norm of the traceless part of the weighted frame operator and explains why this contribution vanishes under irreducible symmetry.

Leaks such as the Panama and Paradise Papers expose large networks of offshore entities, and they invite an obvious question for network learning. Can the relations these structures are built to hide—above all, who beneficially owns what—be recovered from the public part of the leak by link prediction? We argue that the answer is mostly no, and that the analyses which suggest otherwise are measuring the wrong thing. Our main result is a distribution-free identifiability bound. For any recovery rule that respects graph isomorphism, and that covers every topological link-prediction score together with every message-passing graph neural network, a concealed endpoint left isolated in the observed graph is interchangeable with its structural twins, so its hidden edge cannot be recovered above chance. Isolation is only the sharpest case. In general the ceiling on recovery is set by the size of a node's structural-indistinguishability class, for which the Weisfeiler–Leman colour class is a computable stand-in, and node degree is at best a loose proxy. On the full ICIJ Offshore Leaks graph (814K entities and 84K labelled beneficial-owner edges) a classifier-free, degree-controlled probe reproduces an exact $0.5$ floor for isolated owners, who make up $20.6\%$ of all owners, and a trained graph neural network lands on the same floor. Recovery climbs only as structural distinctiveness grows, and the floor reappears in every one of the five leaks. Along the way we document five evaluation traps. Each one makes a bound that cannot be beaten look beaten, and we give a short rule that avoids them. The practical upshot is to redirect effort from the hidden principal, which is close to unrecoverable, toward the machinery of concealment, and to spell out why fusing external data helps far less than one would hope.

Let $F_n$ be uniform on all forests of the simple complete graph $K_n$, with isolated vertices allowed. A conjecture of Kahn and of Winkler, studied by Grimmett and Winkler, asserts that any two distinct edges of any finite graph are negatively correlated under the uniform forest measure. Stark proved this for $G=K_n$ once $n$ is sufficiently large, but did not furnish an explicit threshold. We prove it for every $n\geq 2$, strictly whenever two distinct edges exist. The difficulty is concentrated in the disjoint-edge orbit, whose correlation ratio tends to one. We remove this cancellation before estimating anything: the desired inequality becomes an exact comparison among the first two moments of the component count and the expected sum of squared degrees. When the component count fluctuates, this comparison contains a variance term absent from the fixed-component identities of Tang and Zhang. Component marking, tail elimination, and effective Stirling bounds control the three moments for $n\geq 651$; exact integer recurrences cover the remaining values.

Let $M$ be chosen uniformly from all matchings of a finite linear $k$-uniform hypergraph $H$, and let $\overline{q}(H)$ be the average probability that a vertex is left uncovered. If $H$ has maximum degree $D$ and normalized average degree $\beta=k|E(H)|/(|V(H)|D)$, then, for every fixed $k\geq 2$ and uniformly in the order, $\overline{q}(H) \leq 1-\beta+(\beta k+o_D(1))\log\log D/\log D$. The underlying estimate is the order-uniform count $\log Z(H)\geq (|E(H)|/D)(\log D-(k+o_D(1))\log\log D)$, and it also counts matchings of size $(1-o_D(1))|E(H)|/D$. We prove this by sampling edges, deleting those incident with unusually large sampled degrees, and controlling only the total deleted mass before applying the Molloy-Reed list edge-colouring theorem. For $d$-regular $H$, a quantitative version of the earlier Asratian-Kuzjurin sampling-to-counting route, using the near-perfect-matching theorem of Gould and Kelly, sharpens the coefficient: for every fixed $k\geq 3$, $\sup_H \overline{q}(H) \leq (\max\{3,k-1\}+o_d(1))\log\log d/\log d$. Kahn's stronger pointwise prediction was disproved by Lee for $k\geq 3$. The qualitative averaged conclusion was already recorded by Kahn and Kim, crediting Anders Johansson, and also follows from the Grable-Asratian-Kuzjurin enumeration; neither source states a rate.

We calibrate credit default swaps and index tranches with elastically stopped Lévy processes: each firm defaults when the running supremum of a latent, spectrally positive distress process crosses an independent exponential barrier. This yields a Cox construction with totally inaccessible default times, while retaining the interpretability and explicit formulas of a structural approach. Adding a single common compound Poisson jump factor to every firm's latent driver gives a parsimonious multi-credit model with simultaneous defaults, which is priced by an exact Wiener–Hopf Monte Carlo scheme. Its tractability rests on a single-name result we prove: a finite partial-fraction formula for the Laplace transform of the default probability under phase-type jumps. On daily CDX North American High-Yield and Investment-Grade panels, our drivers attain the lowest out-of-sample errors in a six-model field and reproduce the inverted spread curves of names heading into default, which a Lévy subordinator provably does not. At the index level, the two-parameter dependence structure closes $73\%$ to $89\%$ of the tranche pricing gap left by independent marginals with the dependence parameters frozen, and up to $95\%$ once re-marked to tranche quotes; our framework dominates a single-factor Gaussian copula and the affine intensity benchmark of Duffie–Gârleanu on both indices.

For a centered convex body $K\subset\mathbb{R}^n$, let $\Psi_2(K)$ denote the symmetric convex body whose support function is given by the $\psi_2$-norms of linear functionals on $K$. The recent solution of Milman's problem on the existence of subgaussian directions by Letwin and Mikulincer naturally motivates the study of the geometry of this body. We prove that $\Psi_2(K)$ has bounded volume ratio with respect to the $L_2$-centroid body $Z_2(K)$. In the isotropic case, we also obtain sharp estimates for its mean width and the volume radii of its orthogonal projections, and derive consequences for the existence of subgaussian orthonormal bases. In particular, we construct orthonormal bases with quantitatively controlled subgaussian constants for every isotropic convex body.

The Cifarelli–Regazzini identity lies at the foundation of the theory of the Dirichlet process and has played a central role in Bayesian nonparametric statistics. By providing the generalized Cauchy–Stieltjes transform of linear functionals of the Dirichlet process, together with an analytic inversion, it yields explicit representations for the distribution function and density of the Dirichlet mean. This transform identity was subsequently extended to the Pitman–Yor process, a generalization of the Dirichlet process, leading to an analogous distributional theory for its linear functionals. Nonlinear functionals, by contrast, remain less understood, with results largely confined to a few examples for the Dirichlet process. In this paper, we move beyond linear functionals by considering the variance of the Pitman–Yor process, a genuinely quadratic functional. In particular, we establish a Cifarelli–Regazzini-type identity providing the generalized Cauchy–Stieltjes transform of the Pitman–Yor variance for general base probability measures under suitable integrability assumptions. We then analytically invert this transform under general conditions on the base probability measure, thus obtaining explicit representations for both the distribution function and density. Corresponding formulas are also obtained for the Dirichlet variance, yielding more explicit representations than those currently available in the literature. We also establish a distributional representation of the Pitman–Yor variance in terms of a Dirichlet process whose base probability measure is the joint distribution of the mean and variance of a suitably specified Pitman–Yor process, mirroring the classical within–between decomposition of variance and providing a quadratic counterpart to a known distributional identity for Pitman–Yor means.

This paper analyzes mixing time lower bounds for random walks on sparse, heavy-tailed Random Intersection Graphs. In sparse feature regimes, heavy-tailed feature distributions lead to the formation of peripheral trap – chains of overlapping low-weight feature cliques attached to high-weight hub nodes within the graph's giant component. By modeling escape trajectories from these traps as continuous limit hitting times for reflected Brownian motion, the analysis demonstrates that random walks experience logarithmic squared delays. Consequently, the mixing time is bounded below by $\Omega(\log^2 n)$, and the local total variation distance exhibits non-concentrated decay, formally preventing a sharp cutoff phenomenon.

Let $x_1,\ldots,x_n$ be independent standard Gaussian vectors in $\mathbb{R}^d$. An ellipsoid fit is a matrix $S \succeq 0$ such that $x_i^\top S x_i =d$ for every $i$, so that all the points lie on the boundary of the centered ellipsoid $\{ x : x^\top S x = d\}$. Saunderson, Parrilo and Willsky conjectured that, as $n,d \to \infty$, this semidefinite feasibility problem undergoes a sharp transition at $n \sim d^2/4$. We prove this conjecture. If $\lim \sup n/d^2 = \alpha^* <1/4$, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose $S$ with all eigenvalues in a fixed interval $[\lambda_- , \lambda_+] \subset (0,\infty)$ depending only on $\alpha^*$. Conversely, if $\lim \inf n/d^2 > 1/4$, then, with probability tending to one, no ellipsoid fit exists, without any spectral restriction. Our proof builds on the Gaussian-equivalence framework developed by Bandeira and Maillard (2025) and closes the two gaps left open in their work: establishing exact fitting and removing the operator-norm constraint. On the satisfiable side, the new ingredients are a head-tail decomposition of the dual vector, exact correction of the sparse head constraints, and a Gaussian comparison principle for the low-influence tail. On the unsatisfiable side, we split a candidate into a low-rank spectral head and a Schatten-3 diffuse bulk, Gaussianize the bulk conditionally on the head, and apply a projected Gordon escape argument. The threshold is governed by the statistical dimension $d(d+1)/4$ of the positive semidefinite cone.

We introduce an infinite-dimensional Galves–Löcherbach system with bounded Elephant-type reinforced synaptic interactions. The reinforcement mechanism modifies the probabilities of excitatory and inhibitory synaptic updates while keeping their amplitudes uniformly bounded. We prove non-explosion by means of a Lyapunov estimate and establish a conditional Wasserstein contraction for the membrane-potential dynamics when the coupled systems share the same reinforcement profile. Finally, we derive the corresponding replica mean-field equation under the Poisson hypothesis.