Papers

In this paper, we study maximum weight spanning trees of the random geometric graph (RGG)~\(G\) formed by~\(n\) vertices where each edge is independently either open or closed with a certain probability and is also equipped with an independent random positive weight. We use segmentation and iterative path construction to obtain deviation bounds for the order of growth of the maximum weight of a spanning tree in terms of an inverse of the edge weight complementary cumulative distribution function (ccdf) and also illustrate our results for the special cases of power law and exponential decay. We then use martingale difference methods to individually estimate the variance contribution due to randomness in vertex locations and edge states/weights and determine sufficient conditions for~\(L^2-\)convergence of the maximum weight, appropriately scaled and centred.

In a preceding paper, we used the graph-theoretic linear framework to show how transient properties of continuous-time Markov processes – splitting probabilities and the moments of first-passage time (FPT) distributions – could be expressed as rational algebraic functions of the transition rates, by using spanning forests of the underlying graph. This contrasts with the related rational formulas for steady-state (s.s.) probabilities, which use only spanning trees. The biophysicist Terrell Hill sketched a procedure for calculating mean FPTs and splitting probabilities in terms of the s.s. probabilities of a modified Markov process, thereby converting calculations using ensembles of trajectories to those using a single trajectory. Similarly, Mark Kac showed that the mean recurrence time to a state of a Markov process could be expressed in terms of the s.s. probability of that state. Here, we explore further the relationships between transient and s.s. properties, forests and trees, and ensemble and single-trajectory calculations. We formalise Hill's procedure by introducing a Hill operator, $H_u[G]$, on a graph, $G$, and use it to calculate all moments of the conditional and unconditional FPTs from $u$ as rational functions of the s.s. probabilities of $H_u[G]$, which arise from trees and "exchange factors" which arise from forests. We then combine this with an unravelling operator, $U_v[G]$, to calculate all moments of the recurrence time distribution as rational functions of the s.s. probabilities of $G$ and related exchange factors. Surprisingly, the Hill operator turns out to be a left-inverse to the unravelling operator, suggesting that the algebra of operators on linear framework graphs may be of broader interest. Our results integrate and generalise previously disparate findings into a common repertoire of rational algebraic formulas for FPTs of Markov processes.

In this paper, we consider an optimal stopping problem with infinite horizon, non-negative pay-offs and non-linear evaluations $\rho_{S,\tau}$ indexed by two indices: $S$ and $\tau$, where $S$ is the time of evaluation and $\tau$ is the time when the pay-off is revealed. The agent's stopping strategies are constrained to be in the set of so-called Bermudan stopping times $\Theta$. Under suitable assumptions on the non-linear evaluations $\rho$ and on the pay-off, we show that a dynamic programming principle holds in this framework. We investigate the existence of $\varepsilon$-optimal stopping times, as well as the existence of optimal stopping times. We show that an $\varepsilon$-optimal stopping time exists. We also prove that the first time when the value family hits the pay-off is optimal if and only if it is finite. We also provide Doob's type convergence for non-negative \emph{$(\Theta, \rho)$}-supermartingales in the case where $\rho_{S,\tau}=\rho_S$ depends on the first index only. We provide an example from BSDEs with infinite horizon.

We prove that every embeddable $3\times3$ row-stochastic matrix is a product of at most seven elementary row-stochastic matrices. Frydman's example shows that this bound is sharp. The proof combines Frydman's six-factor criterion and spiral theorem with a reduction to a three-parameter critical family and an analysis of the membership certificates that persist along it.

We determine necessary and sufficient conditions for complete monotonicity of the Srivastava–Tomovski extension, a generalized Mittag–Leffler kernel arising in fractional calculus, while separating the defining generalized-Wright series from its positive-axis realization when the series is not entire. For positive parameters, let \(\Delta=1+\alpha-\kappa\). In the entire regime \(\Delta>0\), \(E_{\alpha,\beta}^{\gamma,\kappa}(-x)\) is completely monotone on \((0,\infty)\) if and only if \(\alpha\leq\kappa\) and \(\kappa\beta\geq\alpha\gamma\). For arbitrary positive parameters, the corresponding positive-axis realization satisfies the same criterion; when \(0<\alpha<\kappa\), it is defined by a second-kind Wright kernel and remains meaningful through the finite-radius and zero-radius phases of the defining series. We also identify the Bernstein measure in every admissible case. In the interior region it is, after normalization, the pushforward under \(t\mapsto t^\kappa\) of a power-biased Wright distribution; at \(\alpha=\kappa\) it becomes a powered beta law for \(\beta>\gamma\) and the atom \(\Gamma(\gamma)^{-1}\delta_1\) for \(\beta=\gamma\). Necessity in the excluded regions follows from a Mellin-transform zero, uniqueness of finite signed Laplace transforms, and moment-support asymptotics. The Prabhakar criterion is recovered by setting \(\kappa=1\).

Let $B^H$ be a one-dimensional fractional Brownian motion with Hurst index $H\in(1/2,1)$. We study the small-mass limit of the kinetic equation \[ dX_t^\mu=Y_t^\mu\,dt,\qquad \mu\,dY_t^\mu=b(X_t^\mu)\,dt-Y_t^\mu\,dt +\sigma(X_t^\mu)\,dB_t^H, \] where the noise coefficient is state dependent and uniformly nondegenerate. The limiting equation is the Young differential equation \[ dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t^H. \] Under uniform ellipticity and strict dissipativity of the transformed drift, we prove that, for every $t_0>0$ and every $\rho<2H-1$, \[ \sup_{t\ge t_0}d_{TV}\big(X_t^\mu,X_t\big) \le C_{t_0,\rho}\mu^\rho . \] The estimate is uniform on the entire half-line. We also construct stationary solutions on a two-sided fractional-noise space and obtain convergence of their one-time marginals in total variation. The exponent $2H-1$ is identified as the natural endpoint: it is generated by the quadratic velocity term after the Lamperti transform, and it degenerates as $H\downarrow1/2$. A nonconstant uniformly elliptic example is included.

For a graph G let F(G) denote the largest order of a regular induced subgraph of G, and let f(n) = min{F(G) : |V(G)| = n}. A problem of Erdos, Fajtlowicz and Staton asks whether f(n)/log n -> infinity. Dyson and McKay have recently proved f(n) <= (sqrt(2e)+o(1)) sqrt(n) via a type-based random model (arXiv:2604.08215). This paper concerns the opposite direction within the type-based family. We conjecture that every model of the family satisfies F(G) >= (sqrt(2e)-o(1)) sqrt(n) asymptotically almost surely, so that sqrt(2e) is the optimal constant obtainable from the family, and we prove the corresponding statement at exponent level – F(G) >= n^{1/2-eps} – conditionally on two explicitly stated hypotheses: a local limit lower bound for inhomogeneous degree sequences, and a correlation estimate at sublinear overlaps. The complementary overlap range, including full overlap, requires no correlation hypothesis. We further record the exact-curvature first-moment computation that independently identifies sqrt(2e), including a uniform trace bound on its determinant correction, and certified exact computations at orders up to 48 consistent with the predicted crossover F ~ min(n^{2/3}, sqrt(n/L)). This version substantially revises v1; see the note in Section 1.

We introduce conditional cylindrical neural networks for approximating functionals of conditional laws in McKean-Vlasov equations with common noise. Fourier moments of the initial law and truncated signatures of the time augmented common noise are mapped by a mixture density network to a Gaussian mixture approximation of the conditional law. A cylindrical neural network then evaluates the target functional through analytic integrals against this predicted measure. Rough path well posedness and stability provide a conditional law map that is continuous in the initial distribution and the rough driver and agrees almost surely with the classical conditional law at the Itô Brownian lift. Combining this continuity with Fourier separation, signature uniqueness, Wasserstein density of Gaussian mixtures, and neural universal approximation, we prove an $L^2$ universal approximation theorem for continuous square integrable functionals. The numerical study implements the resulting two stage procedure on six examples, including non Gaussian initial laws, nonlinear drift, multiplicative common noise, and a two dimensional state. Independent particle references are used when no closed form law is available. The learned conditional law and functional approximations consistently improve on the empirical particle plug in, and additional experiments examine feature sensitivity, training from one terminal observation per common noise scenario, and Itô–Stratonovich consistency.

The shape function of a pair of finite-valued random variables was introduced in arXiv:2606.23849, where it was used to derive a spectral bound on the entanglement of the pair, a quantity measuring the extent to which their mutual information can be extracted. In this article, we further develop the theory of shape functions for pairs of random variables. We prove that, when $(X,Y)$ is uniformly supported on the edges of a biregular bipartite graph, the value of the shape function $\mathcal{S}(X,Y)(\alpha,\beta)$ equals the logarithm of the operator norm of the graph's incidence matrix with respect to Lebesgue exponents determined by $(\alpha,\beta)$.This identification, in particular, enables the numerical approximation of the shape function and, by duality, of the extension profile, also known as the tension region, of the pair. We also establish a collection of relations and inequalities satisfied by shape functions, including convexity and monotonicity properties, composition inequalities, and relations describing their behavior under conditioning and the adjoining of variables.

Molecular motion within living cells provides a window into the physical principles underlying intracellular organization and transport. Yet a fundamental limitation remains: measured trajectories are often short, noisy, and randomly oriented, rendering spatial anisotropies inaccessible to conventional analyses. We show that turning-angle statistics provide a robust approach for uncovering anisotropies in anomalous diffusion. Using a two-dimensional anisotropic fractional Brownian motion model with random orientations, we show theoretically and through simulations that turning-angle distributions preserve signatures of anisotropy. We apply this approach to single-particle tracking data, including quantum dots in the cytoplasm of HeLa cells and membrane proteins in hippocampal neurons. Turning angles reveal hidden anisotropies in the motion of quantum dots and Nav 1.6 channels, and in specific dynamical states of glycoprotein CD4. These results establish turning-angle analysis as a powerful strategy for detecting organization in complex environments and reveal that anisotropic anomalous diffusion is an overlooked feature of intracellular dynamics.

We study random Bernoulli measures $\mu_{\mathbf{p}}$ in a random environment $\mathbf{p}=(p_1,p_2,\ldots)$, where $p_n$ are independent and identically distributed on $(0,1)$. For almost every environment $\mathbf{p}$, we establish several almost sure properties of $\mu_{\mathbf{p}}$: its local dimension, $L^q$ dimensions, Rajchman property, mutual singularity with any fixed measure, and a dichotomy for normal numbers of its typical points.

We establish functional quantization and empirical Wasserstein rates for continuous Itô processes under the supremum norm. We assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>1$. Under this assumption, the $n$-point $L^q$ quantization error is at most $C(\log n)^{-1/2}$ for every $1\leq q<\rho$. The known Brownian lower bound shows that the exponent $1/2$ is optimal over this class. Our proof combines an adaptive dyadic time partition with localization according to the size of the integrands. A general transfer principle yields the sharp mean rate $(\log N)^{-1/2}$ for the $p$-Wasserstein distance between the empirical law of $N$ independent copies and their common path law, together with nonasymptotic deviation bounds, whenever $1\leq p<\rho$. Applications include empirical path-law estimates for path-dependent SDEs and a path-space limit-theory estimate for path-dependent McKean–Vlasov interacting particle systems with common noise.

Altschuler and Chewi conjecture that the sub-Gaussian Orlicz–Wasserstein shifted Rényi divergence between two isotropic Gaussians of equal covariance is attained by a deterministic shift of the mean, which would give an exact closed form for the shift budget consumed by their analysis. In this note, we show that this conjecture is true when $q=1$, where the Rényi divergence degenerates to the Kullback–Leibler divergence, and false for every $q>1$ outside the degenerate regime.

We give a new proof by flow decomposition of the known Grigor'yan type integral criterion for stochastic completeness of weighted graphs. The main ingredient is a finite-domain estimate relating the mass of a killed $1$-resolvent and the current reaching the boundary to relative capacities of intrinsic balls. This estimate yields logarithmic criteria in terms of capacity to infinity and relative capacity, and recovers the volume criterion by an intrinsic cutoff, without passing to a metric graph or refining the graph.

We prove a derivative-free analytic linear spectral statistics theorem for entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The transform is required to be centered, variance-normalized, and admissible under the small translations generated by the spike: its shifted mean and second moment have first- and second-order profiles, while its centered shifted fourth cumulants and fourth tails are stable. At every microscopic shift we construct an explicit uniformly bounded three-point variable matching the first four centered moments of the target entry exactly. A generalized-Wigner LSS theorem applies to the resulting bounded triangular array, and a global Fourier–Duhamel derivative estimate transfers the analytic statistic back to the rough transform without a common truncation, coefficient-stability assumption, or local law. The order-one mean consists of the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. The centered covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter $\kappa_4^{f,\nu}$. We distinguish the bulk contour statistic from the full trace in the supercritical regime and show that the separated outlier adds exactly $\varphi(\theta+\theta^{-1})$ to the full-trace centering. As a self-contained consequence, Gaussian noise with $f\in L^{4+\epsilon}(\gamma)$ satisfies the theorem without differentiability of $f$; the resulting bulk and full-trace corollary has explicit Hermite coefficients and includes every centered, variance-normalized polynomial-growth transform. Concrete likelihood-ratio, smooth-transform, bounded rough-transform, and atomic criteria are provided, together with obstructions showing that bare $L^4(\nu)$ is insufficient.

In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller–Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya–Babuška–Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller–Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.

Hawkes-based microstructural foundations for rough volatility, leverage, and rough Heston-type limits were developed by El Euch et al. (2018, Finance Stoch., 22(2), 241–280) and connected to the affine rough Heston framework of El Euch and Rosenbaum (2019, Math. Finance, 29(1), 3–38). The rough Hawkes–Heston model with common price–volatility jumps of Bondi et al. (2024, Math. Finance, 34(4), 1197–1241) extends this framework by adding state-dependent common jumps to rough affine volatility. We provide a microstructural foundation for its variance and common-jump mechanism by constructing a Poisson-embedded marked Hawkes order-flow model. Ordinary arrivals generate rough continuous volatility and leverage through a nearly unstable heavy-tailed Hawkes mechanism, while rare marked arrivals represent common shock events that produce simultaneous price jumps and volatility excitation. Under the nearly unstable scaling and the reduced-form admissibility conditions, the complete rescaled price/variance/jump system converges along the full sequence to the unique complete canonical rough Hawkes–Heston weak solution. The Hawkes renewal structure yields a Mittag–Leffler Volterra representation, which is then rewritten in Riemann–Liouville fractional form. The limiting coefficients are expressed explicitly in terms of the microscopic parameters. The construction provides a microstructural foundation for the variance and common-jump mechanism of the rough Hawkes–Heston model. Numerical experiments illustrate the convergence of our microstructural foundation to the rough Hawkes-Heston model.

This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\omega)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,\omega)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,\omega)$.

Stochastic gradient descent for a loss function discontinuous across lower dimensional manifolds is analyzed by studying its differential equation limit.

Let $\xi$ be a centered real subgaussian random variable with positive variance and a bounded Lebesgue density, and let $A_m\in\mathbb{R}^{N_m\times m}$ have independent entries distributed as $\xi$, where $N_m/m\to\gamma>1$. For each set $I\subset[N_m]$ with $|I|=m$, let $(A_m)_I$ denote the row submatrix indexed by $I$, and define $M_m(A_m):=\min_{I\subset[N_m],\,|I|=m}\sigma_{\min}((A_m)_I)$. We determine its exponential scale: $\frac{1}{m}\log M_m(A_m)\xrightarrow{\mathbb{P}}-h(\gamma)$, where $h(\gamma):=\gamma\log\gamma-(\gamma-1)\log(\gamma-1)$. This extends the corresponding real Gaussian result. The main new ingredient is an upper-tail argument that avoids uniform control over exponentially many random hyperplanes. We combine a density-level local central limit theorem for delocalized directions, an averaged delocalization estimate for hyperplane normals, an exponential bound for nearly parallel pairs, and amplification using a linear number of independent probe rows. For every fixed $\varepsilon\in(0,h(\gamma))$, the probability of an $\varepsilon$-deviation is at most $C\exp(-c\sqrt{m})$ for all sufficiently large $m$. Under the canonical coupling induced by a single infinite i.i.d. array, this summable deviation estimate yields a uniform almost-sure exponential law over every compact range of aspect ratios. In particular, at the real phase-retrieval threshold $N_m=2m-1$, the Balan–Wang stability parameter has exponential base $1/4$ in probability and, under this coupling, almost surely.

We prove a version of Föllmer's pathwise Itô formula for weakly differentiable functions of continuous paths with finite quadratic variation along a sequence of partitions. For each such path $\omega$, we introduce a path-dependent Sobolev space $W_{\omega,\pi}^{2}$ defined through smooth approximation of the weak Hessian in a seminorm generated by discrete weighted occupation measures of the path $\omega$. For $F\in W_{\omega,\pi}^{2}$, we construct the pathwise integral $\int \nabla F(\omega)\,d^{\pi}\omega$ and the covariation $[\nabla F(\omega),\omega]_{\pi}$, and prove the change-of-variable formula $$ F(\omega(t))-F(\omega(0)) = \int_{0}^{t}\nabla F(\omega(s))\,d^{\pi}\omega(s) + \frac12[\nabla F(\omega),\omega]_{\pi}(t). $$ Our result does not require any assumption on the existence of local time for the path; the Ito term appears as a quadratic covariation. For Brownian motion, we show that functions in $W^{2+,p}(\mathbb{R}^{d})\cap W^{2,1}(\mathbb{R}^{d})$ belong almost surely to the corresponding path-dependent space, outside a polar exceptional set of starting points. If, in addition, $F\in W_{\mathrm{loc}}^{1,2}(\mathbb{R}^{d})$, the pathwise integral agrees with the stochastic Itô integral, yielding a pathwise version of the multidimensional Föllmer–Protter formula.

We develop a unified framework for constructing combinatorial structures under local constraints. Our approach extends the configuration model for random graphs with a prescribed degree sequence, and covers many special cases, including bipartite graphs, directed graphs, oriented graphs, edge-colored (bipartite) graphs, and (directed) hypergraphs. By reformulating half-edge matching as an independent set problem in an auxiliary graph, we identify 2-uniformity, a property characterising when greedy sampling preserves asymptotic uniformity. We classify all 2-uniform graphs and show that only two classes, the configuration space and the bipartite configuration space, have unbounded independence number, enabling the asymptotic regime. Our main theorem then gives the asymptotic sampling distribution and enumeration formulae for configurations, with error terms of order $O(d_{\max}^4\log m/m+d_{\max}^2(\log m)^2/m)$ as the number of edges $m$ tends to infinity with maximum degree $d_{\max}=O(m^{1/4}/\log m)$. This settles the long-standing $O(m^{1/4-\tau})$ bound (for some fixed $\tau > 0$), making the critical exponent explicit. Furthermore, our theorem accommodates forbidden edges, provided that each vertex participates in at most $O(m^{1/4}/\log m)$ of them. In particular, this enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.

One hallmark of exactly solvable KPZ random growth models is product-form invariant measures. In the setting of exponential last-passage percolation (LPP), this corresponds to the independence of Busemann increments along any down-right path. However, this independence breaks down when multiple asymptotic directions are considered simultaneously, owing to the fact that jointly invariant measures are not jointly product-form. This paper shows that the failure of independence is one-sided: Busemann increments across arbitrary directions are negatively associated. As an application, we derive an exponential concentration inequality for sums of Busemann increments on the diffusive scale, even when the increments are not independent. While our argument relies on a Burke property that is special to exponential weights, all other proof ingredients$\unicode{x2014}$including hidden LPP monotonicities and braid relations for queueing maps$\unicode{x2014}$hold for arbitrary weights.

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

Let $\mathrm{CHL}_N$ be the cylindrical Hastings–Levitov aggregation process with parameter $0$ on a cylinder of width $N$ with particles of fixed size $\lambda>0$, and let $\omega_{N,\lambda}$ be its tree-completion time — the last time at which a new tree is born on the base circle. Chen, Procaccia and Zong proved the sharp upper bound $\mathbb{E}[\omega_{N,\lambda}]\le(1+\varepsilon)(\log N)/(2\lambda)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \[ \lim_{N\to\infty}\frac{\mathbb{E}[\omega_{N,\lambda}]}{\log N}=\frac{1}{2\lambda} \qquad\text{for every fixed }\lambda>0 . \]

Let $G$ be a finite connected multigraph whose edges receive independent weights from one atomless law, and let $\operatorname{MST}(G)$ be the resulting random minimum spanning tree. Its law is not pairwise negatively correlated: Lyons, Peres and Schramm exhibited two positively correlated edges, and we give such an example on a simple graph. We prove that positive correlation is nevertheless uniformly controlled: $\mathbf{P}(e,f\in T)\leq 8\mathbf{P}(e\in T)\mathbf{P}(f\in T)$, answering a question of R. Lyons recorded by Tang and Zhang. After conditioning on all other weights, Harris's inequality gives conditional negative correlation; two bottleneck distances and a sharp second-moment estimate control the remaining environmental covariance. For $K_n$ we prove pairwise negative correlation for every $n\geq 3$. The key finite identity is $\mathbf{E}[\mathrm{deg}(x)^2]=10(n-1)/n-4\mathbf{E}[L_n]$, where $L_n$ is the total weight of the minimum spanning tree under rate-one exponential weights. Known expansions for $\mathbf{E}[L_n]$ then give the rate of convergence to $10-4\zeta(3)$ and the limits of both pair-correlation ratios. Finally, an explicit $K_4$ family shows that no universal constant survives when the independent edge laws need not be identical.

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially given an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. We also consider the offline version of the process in which the algorithm is given the entire sequence of random vertex choices before the selection takes place. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. In this paper, we focus on the property of being triangle-saturated and establish upper and lower bounds on the number of rounds required to construct a triangle-saturated graph in both the online and offline versions of the process.

A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussian projections and Haar-random Stiefel projections have exactly the same knot-type law, despite having different metric shapes, and that at every stick budget the model gives positive probability to precisely the knot types realizable with that many sticks. Its linear-algebraic structure yields an exact marginal distance law, an exact mean planar-crossing count, and crossing concentration, while in the first nontrivial six-stick case the complete tetrahedral sign pattern gives an exact unknot-versus-handed-trefoil classifier for every generic sample. The result is a direct bridge from random projections and finite sign geometry to the topology of random knots.