Papers

Autonomous small-body proximity operations require continuous-thrust guidance under uncertain gravity, solar-radiation pressure (SRP), and stochastic disturbances. We combine iterative covariance steering (iCS) with $L_1$ distributionally robust adaptive control ($L_1$-DRAC) to provide risk-aware collision avoidance despite mismatch between the nominal planning model and the true dynamics. Iterative covariance steering plans a nominal feedback policy satisfying distributionally robust conditional value-at-risk keep-out constraints, while $L_1$-DRAC compensates for dynamics uncertainty and bounds the residual distributional mismatch through a Wasserstein radius used by the planner. In $\mathbf{1000}$ paired Monte Carlo trials with gravity, SRP, and diffusion mismatch, $L_1$-DRAC adaptive control reduces the keep-out violation rate from $\mathbf{32.7\%}$ to $\mathbf{0.2\%}$ and the physical collision rate from $\mathbf{16.4\%}$ to zero, without actuator saturation.

We study testing a random geometric graph with a fixed kernel against an Erdős-Rényi graph with the same edge density. Under the null hypothesis, the graph has $n$ vertices and independent edges with probability $p$. Under the alternative hypothesis, latent points $X_1,\ldots,X_n$ are independent and uniform on $\mathbb{S}^{d-1}$, and edges are conditionally independent with probabilities $K(\langle X_i,X_j\rangle)$, where $p=\mathbb{E}[K(\langle X_1,X_2\rangle)]$. The goal is to determine a critical dimension $d_\star = d_\star(n,K)$: the total variation distance between the two graph laws tends to one for $d\ll d_\star$ and to zero for $d\gg d_\star$. For dense graphs, [BDER16] established $d_\star\asymp n^3$ for hard threshold kernels, whereas [MWX26] proved $d_\star\asymp n^{3/4}$ for smooth kernels with $K'(0)\ne0$. We connect these results by characterizing the detection thresholds for kernels that need not be differentiable at zero. We also determine the thresholds for smooth kernels whose first nonzero derivative at zero has finite order greater than one. Throughout these classes, signed triangle or quadrilateral counts attain the optimal detection threshold. For certain nondifferentiable kernels, quadrilateral counts are optimal while triangle counts are suboptimal.

We establish a nonasymptotic Gaussian approximation bound over centered Euclidean balls for sums of independent, not necessarily identically distributed, random vectors in \(\mathbb R^p\). Let \(S_n\) denote the sum, let \(\Sigma=\operatorname{Var}(S_n)\), and let \(G\sim N(0,\Sigma)\). Define the effective rank and the normalized squared spectral mass by \[ r(\Sigma)=\frac{\operatorname{tr}\Sigma}{\|\Sigma\|}, \qquad v(\Sigma)=\frac{\operatorname{tr}(\Sigma^2)}{\|\Sigma\|^2}. \] Under a uniform directional fourth-moment condition, covariance-balance assumptions, and the spectral conditions \(r(\Sigma)\le C_\Sigma v(\Sigma)\) and \(v(\Sigma)\ge6\), we prove \[ \sup_{z\ge0} \left| \mathbb P\{\|S_n\|^2\le z\} - \mathbb P\{\|G\|^2\le z\} \right| \le C\,\frac{r(\Sigma)^{3/2}}{n}. \] The constant depends only on the fixed parameters in the assumptions and is independent of the ambient dimension and individual small eigenvalues of the covariance matrix. The bound is uniform at the endpoint \(v(\Sigma)=6\), without a logarithmic loss. Its main feature is the combination of the \(n^{-1}\) rate with dependence on effective rank, including this endpoint. Consequently, the bound can vanish when \(r(\Sigma)=o(n^{2/3})\), even when \(p\) is much larger than \(n\). The same estimate extends to random elements of a separable Hilbert space with trace class covariance. The proof builds on the Fourier and symmetrization methods of \citet{GoetzeZaitsev2014}, combined with moment matching, truncation, and comparison estimates that retain the covariance structure.

Hybrid-abandonment systems, where jobs abandon from certain stages of service but not others, occur frequently in medical, service, and computing systems. Abandonment affects the heavy-traffic behavior of both the stages with and without abandonment. In contrast to existing heavy-traffic works that focus on cases where there is abandonment at either all stages or no stages, we focus on the heavy-traffic behavior for these hybrid-abandonment systems. We provide the first heavy-traffic characterization of the second queue length in a two-stage tandem queue with abandonment only in the first stage. We characterize the scaled mean of the second stage queue length and prove that the scaled second queue length is asymptotically exponentially distributed and asymptotically independent of the first stage queue length. We derive our results by analyzing Markovian arrival process queues with countably infinite phases. We do this by extending techniques used to analyze queues with Markov-modulated arrivals to the broader class of infinite Markovian arrival process queues. We prove asymptotic results for a more general infinite phase process condition, an infinite birth-death phase process, and the hybrid-abandonment tandem case. We also empirically demonstrate that our heavy-traffic approximations are good approximations across various system settings.

We introduce a new stochastic model of telomere length dynamics during hematopoiesis. The numerous cell divisions occurring in this setting makes telomere length dynamics, and in particular their attrition, a crucial aspect of hematopoiesis. After recalling the main aspect of the biological setting, we describe an individual based model of hematopoietic population cells. We then study its behaviour in the large population limit, as well as its fluctuations.

Reversible motion can produce a directed circulation of surviving ancestry in a \FV\ particle system. We prove that two particles are the exact threshold for universal ancestral balance in finite systems with uniform copying. For two particles, the stationary current of surviving labels vanishes for every finite irreducible reversible Markov chain and every positive killing vector. On a fixed complete three-state graph, nonzero circulation occurs for each population size at least three on an open dense set of killing vectors of full Lebesgue measure. A single killing vector produces circulation simultaneously for every population size at least three, and such vectors form a dense $G_\delta$ set of full measure. The current is divergence free and vanishes on bridges, so three sites and three particles are the smallest possible setting for circulation. We also identify the probabilistic effect of this circulation: for sufficiently rare transplants, migration followed by fixation is more likely in one direction than in the reverse direction.

It is conjectured that in the Lorentz mirror model on $\Z^2$ all trajectories are finite. Quas showed that, almost surely, the two halves of any infinite trajectory, cut at an edge, meet at infinitely many vertices, each of which is pivotal: changing its state can make the trajectory finite. We construct plane graphs on which infinite trajectories exist, and all of them have this property. Given $p\in(0,1)$, we replace each edge of a random one-ended subtree of $\Z^2$ by a number of parallel edges, depending on $p$, that is at most logarithmic in the height of the corresponding descendant tree. The medial graph of the resulting plane multigraph is a factor of the tree, and for the uniform spanning tree it has finite vertex intensity. In the mirror model on this medial graph, with probability $p$ for mirrors along primal edges and any probability $r\in(0,1-p]$ for mirrors along dual edges, infinite trajectories exist, and the two halves of each of them meet infinitely often. The construction rests on Bernoulli percolation: every random one-ended locally finite tree can be decorated in this way such that bond percolation with parameter $p$ is critical and percolates, and every vertex of the infinite cluster has infinitely many pivotal edges. We characterize this property for bond and site percolation and for mirrors.

For $\Delta\ge2$ and $q\ge11\Delta/6$, we prove that the antiferromagnetic $q$-state Potts partition function on finite simple graphs of maximum degree at most $\Delta$ has no Fisher zeros in a graph-uniform complex neighbourhood of $[0,1]$. For $q>11\Delta/6$, the proof establishes coupling independence throughout $[0,1]$ using a soft version of Vigoda's flip dynamics. Our main tool is a separator-shell transfer theorem for the Potts model on induced-subgraph closed classes of graphs of maximum degree at most $\Delta$, with $q\ge\Delta+1$. It yields a graph-uniform zero-free neighbourhood of $[0,1]$ from Hamming coupling independence at $0$ and a uniform coupling-independence bound on each interval $[\delta,1]$, $\delta\in(0,1]$. We also obtain zero-free Lee-Yang polydiscs around the uniform field for vertex- and edge-colour fields. For Boolean Holant problems on bounded-degree graphs whose signatures come from a fixed finite family of log-concave symmetric signatures $f$ with $f(0)>0$, such as $b$-matchings, we obtain graph-uniform zero-free polytubes around every bounded box of nonnegative activities; their union is an open zero-free neighbourhood of the nonnegative orthant. An appendix summarizes further coupling-independence inputs and the zero-free regions they yield.

In this note we prove in a broad analytic set-up a general result characterizing the asymptotic vanishing of the $L^2$-relaxation time of averages in the presence of a large incompressible drift-like term. The broad analytic set-up is analogous to that of Constantin-Kiselev-Ryzhik-Zlatos '08 or of Coti Zelati-Delgadino-Elgindi '20. A crucial role is played by the singular value gap method of Chatterjee '25, further developed by Eberle-Lörler '26.

While models of undirected and directed geometry in random i.i.d. disorder with rapidly decaying tails are expected to behave similarly and exhibit features of the Kardar–Parisi–Zhang universality class, the scenario when the background noise is fractal is expected to be significantly different. A canonical example of the latter is Liouville Quantum Gravity, where the planar Gaussian free field forms the fractal background. A toy model capturing some of the essential features is given by Mandelbrot's fractal percolation, a random Cantor set. In the simplest setting, dyadic cubes of different scales are retained independently with probability $p$ (a parameter of the model), and the intersection of all retained cubes forms the set of open sites. Connectivity properties of such sets have been intensely studied. Across [CCD88, Cha95], a surprising result was proven. Namely, while there is a phase transition for connectivity of the fractal percolation set (like in usual bond percolation), directed percolation never occurs. Results of a similar spirit have been obtained in geometric measure theory as well. As a step towards understanding last passage percolation (LPP) driven by the Gaussian free field (initiated in [GGN24]), we study the problem of quantifying LPP in fractal percolation. Developing a multi-scale framework, we show an almost polynomial gap from linear growth of the passage time. The obstruction to high LPP values is the Lipschitz nature of directed paths. Along these lines, [Cha96] showed that any connected subset of fractal percolation must have Hausdorff dimension strictly larger than one. We show that one can construct directed paths passing through open sites in fractal percolation provided they move exponentially fast in the spatial direction—a model we introduce and term as unrectifiable directed percolation.

We study Gaussian regression under squared population $L_2$ loss in a known $m$-dimensional subspace of degree-at-most-$k$ functions on the $d$-dimensional Boolean cube. Random inputs can undersample regions essential for prediction, delaying the parametric rate even when the model is known. For fixed $q_0<1/2$, $1\le k\le q_0d$, and sufficiently large fixed $A$, the worst-subspace sample threshold for minimax error $A\sigma^2(m+t)/n$ with confidence $1-e^{-t}$, $t\ge\log4$, is \[ N=(m+t)\exp\{E_{d,k}+O(k^{1/3})\}, \quad E_{d,k}=d\Psi(k/d), \] where $\Psi(q)=\log2-\mathsf H(\tfrac12-\sqrt{q(1-q)})$ and $\mathsf H$ is binary entropy with natural logarithms. The upper bound holds for every feasible $m$; the matching lower bound holds when $m\le\binom d{\lfloor k^{1/3}\rfloor}$ or $t\ge m$. We sharpen the Polyanskiy–Samorodnitsky uncertainty principle in two respects. First, for fixed leakage $\rho\in(0,1)$, the smallest set carrying a fraction $1-\rho$ of a nonzero degree-at-most-$k$ polynomial's energy has probability $\exp\{-E_{d,k}+O_{\rho,q_0}(k^{1/3})\}$. An Airy-kernel construction proves that the remainder cannot be $o(k^{1/3})$ in general. Second, we construct a subspace of dimension $\binom d{\lfloor k^{1/3}\rfloor}$ such that every function in the subspace has at least a fraction $1-\rho$ of its energy on the same set, whose probability is at most $\exp\{-E_{d,k}+C_{\rho,q_0}k^{1/3}\}$. For sufficiently large $k$, this set is a Hamming ball. A striking consequence is an exponential cost of noise: the parametric rate can require $(m+t)4^k\exp\{-O(k^{1/3})\}$ samples, whereas $O((m+t)2^k)$ suffice for noiseless identification. As $k\to\infty$ with $k/d\to0$, the noisy threshold is $(m+t)\exp\{2k+o(k)\}$.

We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional $Z_T=\int_0^T W_s^2ds$, where $(W_t)_{t\geq 0}$ is a Brownian motion with drift $\mu$. More precisely, we consider the conditional law of $W$ given the event $Z_T\le \theta T$, as $T\to\infty$, allowing the drift $\mu$ to depend on $T$. We derive sharp, uniform small-deviation asymptotics for $Z_T$, including the exact prefactor, and use them to show that the conditioned process converges weakly to an Ornstein–Uhlenbeck process. Remarkably, the limiting dynamics are independent of the original drift.

We study a Brownian particle in $\mathbb{R}^d$, $d\geq2$, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The proof uses a renormalization group induction: at each scale, the elliptic operator is well approximated by a Laplacian with effective diffusivity decaying as a power of scale. We compute this exponent exactly in dimension two.

Large-scale information retrieval systems, including retrieval-augmented generation (RAG) and recommendation engines, widely use multi-layered hierarchical data structures for ultra-fast approximate nearest-neighbor search in high-dimensional vector spaces. However, the geometric conditions that ensure accurate and efficient greedy navigation remain poorly understood. In this work, we study the efficiency of greedy navigation on a hierarchy of proximity graphs constructed from \(n\) data points on the \(d\)-dimensional torus~$\mathbb{T}^d$. We identify a deterministic coverage condition under which, given any query $q\in \mathbb{T}^d$, greedy search returns a point within $(1+\varepsilon)$-factor of the distance to the closest point. This coverage property holds with high probability when the data is distributed as a homogeneous Poisson process, a Hermitian determinantal process, or a bounded-density Cox process, as long as $d = o(\log n/\log \log n)$. Under the same assumptions, the expected number of greedy hops for a fixed query is \(O\!\left(\exp\!\left(\tfrac12 d\log d+O(d)\right)\log n\right)\), yielding logarithmic expected hop count in fixed dimension.

Multiparameter complex processes, based on the Costa-Farber model, provide a common framework for growing random simplicial complexes, including the Linial-Meshulam and random flag complex processes. We establish laws of large numbers for persistent Betti numbers of these processes and prove that, after a deterministic time change and normalization, their persistence diagrams converge weakly in probability to a deterministic finite measure. Its quadrant masses are determined by universal limits of normalized persistent Betti numbers, characterized by an explicit finite-dimensional optimization problem. We extend the diagram convergence to integrals of continuous functions with polynomial growth in the death coordinate and controlled singularities at the diagonal. These results yield convergence of additive and multiplicative lifetime measures and total persistence. We further deduce convergence in optimal transport distance. The proofs combine local weak convergence of simplicial pairs, spectral analysis of compressed Laplacians accounting for dependence through shared faces, and quantitative estimates for large death coordinates and short lifetimes.

For a finite set $A\subseteq\mathbb Z^d$, we study the capacities of a sausage obtained by transporting $A$ along the Minkowski sum of simple or critical branching random walks. We first establish quantitative capacity estimates for the random walk sausage and the branching sausage. This in turn, gives us estimates for the probability that Minkowski sums of a mixture of walks hit a finite set in high dimensions. Our approach is based on an annular decomposition of a critical branching random walk into successive waves, which may have further applications.

We consider a once-excited random walk on $\mathbb{Z}^d$, $d \ge 2$, where the walk on its first visit has a bias of constant strength $\beta>0$ toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after $n$ steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to $n^{1/(d+1)}$, and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.

We determine the percolation threshold of the Brownian loop soup on a transient locally finite weighted tree with proper cable metric and no killing. It equals $1-D_G/2$, where $D_G\in[0,2]$ is the Hausdorff dimension of the boundary in the metric induced by the normalized Green function. This gives an analogue of Lyons' branching-number formula for Bernoulli percolation, with Green distance replacing graph distance. Spherically symmetric trees with unit conductances and branching number two realize every threshold in $[1/2,1]$, while geometric conductances give explicit positive thresholds below one-half and a subdivided binary tree has threshold zero. Under the capacity assumption of Drewitz, Prévost and Rodriguez, we characterize the threshold one-half by a finite-energy condition, without a minimum-degree assumption. Non-percolation at one-half then follows from their theorem and Lupu's coupling. For supercritical Galton-Watson trees with unit conductances, conditioned on survival, the boundary dimension is almost surely one and the threshold is one-half, with non-percolation at criticality. No offspring moment assumption is required, and leaves and vertices with one child are allowed. The classification also applies intrinsically to proper real trees spanned by their infinite rays.

Random-walk loop-catchers (RWLC) are a one-parameter family of random subsets of the trace of a random walk, recently introduced in [Cai 2026, arXiv:2607.18070], which interpolates between loop-erased random walk (LERW) and the full walk trace, satisfying a recovery property. We prove that the continuous version of this recovery property uniquely characterizes the scaling limit of LERW and RWLC in three dimensions, using a Green-function test via entangled multipath LERW, as outlined in [Cai 2026, arXiv:2607.18070]. For LERW, this resolves the 3D case of the conjecture of Sapozhnikov–Shiraishi [Sapozhnikov–Shiraishi 2018, Probab. Theory Related Fields 172, 615–662] and gives a new and axiomatic proof of the existence of the scaling limit, which was first proved in [Kozma 2007, Acta Math. 199, 29–152]. For RWLC, this gives the construction of 3D Brownian loop-catchers. The characterization also yields inversion invariance of the 3D LERW scaling limit.

We adapt the 'induction over scales' scheme of Duminil-Copin, Manolescu, and Tassion to the FK-Ising model on s-embeddings. Namely, we prove the existence of crossings of rough topological quadrilaterals with unfavorable boundary conditions, known as 'strong' RSW, under the 'uniformly bounded geometry' assumption (all edge lengths are comparable to 1 and all angles are bounded from below) on the tangential quads that form an s-embedding. This result has a number of standard consequences like quasi-multiplicativity of arm events and fractal properties of the clusters. It can be applied in a variety of setups including near-critical Ising models on regular grids below the correlation length scale by considering appropriate s-embedding thereof.

From previous results it follows that probability distributions on metric spaces featuring unique Fréchet means are dense among measures admitting means in both the quadratic Wasserstein metric and the total variation metric. Additionally, we show that the converse also holds on complete finite-dimensional noncontractible Alexandrov spaces with curvatures bounded from below, which encompass compact manifolds without boundaries and nonmanifold shape spaces: Probability distributions featuring nonunique Fréchet means are also dense in both the Wasserstein metric and the total variation metric. Moreover, in either metric, no nonempty open set of probability measures admits a continuous selection of means. This sharpens a previous result on zero reach of the Dirac embedding. Our argument combines cut-locus avoidance for atoms with a topological obstruction. For Riemannian manifolds, all of our conclusions hold for all exponents $1 < p < \infty$, also.

We study the transient behavior of two threshold-dependent stochastic models: a Cramér–Lundberg risk model and an M/G/1-type queueing model. In both models, the dynamics are driven by two different compound Poisson processes with drift, with the governing process depending on whether the current state is below or above a fixed threshold. For the risk model, we characterize the probability of ruin before an exponentially distributed epoch, while for the queueing model, we characterize the Laplace–Stieltjes transform of the workload at an exponentially distributed epoch. These quantities are characterized using fluctuation-theoretic results for spectrally positive Lévy processes, with Laplace transforms taken with respect to time for the risk model and with respect to both state and time for the queueing model. We consider general claim-size and job-size distributions and provide explicit representations of the relevant auxiliary quantities when these distributions are of phase-type.

In this paper we consider random walks moving in a dynamic random environment that is completely resampled with probability $p$ after each step of the random walker. This resampling yields a straightforward regeneration structure and limit theorems, such as a law of large numbers with a limiting speed $v(p)$ that depends on $p$. We investigate several properties of $v(p)$ as a function of $p$, such as differentiability and monotonicity. Similarily, we study the limiting diffusivity $\sigma^2(p)$ in a central limit theorem.

Let $X_1,\ldots,X_n$ be almost surely distinct independent real-valued random variables. Let $\sigma$ be the random permutation of $\{1,\ldots,n\}$ such that $X_{\sigma(1)}<X_{\sigma(2)}<\cdots< X_{\sigma(n)}$. We show that the set of laws of $\sigma$, as the laws of $X_1,\ldots,X_n$ vary, has semialgebraic dimension \[ \sum_{k=2}^n {n \choose k}(k-1)! \] as a subset of the $(n!-1)$-simplex. This establishes a conjecture of Babson, Duchin, Iseli, Poggi-Corradini, Thurston, and Tucker-Foltz who proved that the dimension is upper bounded by the above expression. We prove their conjecture by constructing a family of finitely supported laws for $X_1,\ldots,X_n$ that provides the correct dimension. We also give an alternative proof of the upper bound using a theorem of Radford. Furthermore, we show that the Mallows law on permutations cannot arise as a law of $\sigma$ for $n\geq 4$, answering a question of Ed Crane.

We study bond percolation on the one-dimensional lattice in which two sites at distance $r$ are connected with probability $p_r=C/r^{1+\sigma}$, where $0 < \sigma < 1$. An infinite cluster exists if $C$ exceeds a critical value $C_c(\sigma)$, which is not known exactly: the available results are the lower bound of Schulman, $C_c\ge1/[2\zeta(1+\sigma)]$, and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives $C_c\le\sigma+O(\sigma^2)$ as $\sigma\to0$ and is below $1$ for $\sigma < 0.667$. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: $C_c^{\rightarrow}\zeta(1+\sigma)\to1$ as $\sigma\to0$. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound $C_c < 1$ for every $\sigma < 1$. We compare the bounds with the known numerical values of $C_c$.

We represent (weighted-)selection-elliptical distributions as an affine combination of the $q$ selection variables plus an elliptical term whose direction alone is independent. This form suffices for $q+2$ fund separation via first-order stochastic dominance, inter alia relaxing Simaan's (1993) three-fund assumptions.

We consider long-range percolation (LRP) on $\mathbb{Z}^d, d \ge 1$ with connection probabilities $\mathbf{p}(x, y) \approx \frac{\beta}{\|x-y\|^s}$ for $\beta>0$ and $s > \min(2d, d+2)$. Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length $n$ is of order $n^2$. To establish the upper bound, we show that a $d$-dimensional isoperimetric inequality holds almost surely simultaneously for all connected sets inside the giant component that are large enough.

We introduce a new and tractable class of non-Gaussian isotropic random fields on the sphere, which we call, for brevity, Grey Spherical Random Fields (GSRFs). The proposed framework provides a flexible extension of the classical Gaussian theory, while preserving strong isotropy and embedding Gaussian random fields as a limiting special case. The construction is based on random spherical harmonic coefficients obtained as Gaussian mixtures driven by multivariate subordinators evaluated at independent random times (with Mittag-Leffler or more general distribution). This representation allows us to derive explicit characteristic functions, mixed moments and cumulants of arbitrary order. We also investigate asymptotic high-frequency behaviour and identify general conditions under which suitably normalized spherical harmonic coefficients converge to Gaussian scale mixtures, recovering several important models, including stable, Gamma and inverse Gaussian cases, as particular examples. Proving that GSRFs emerge as scaling limits of multivariate continuous-time random walks, with heavy-tailed waiting times, provides a microscopic foundation for the proposed models. We further obtain quantitative bounds measuring the deviation from Gaussianity of the spherical harmonic coefficients. The resulting framework substantially broadens the class of analytically tractable isotropic random fields on the sphere, while preserving explicit probabilistic representations and flexible dependence structures.

We consider weakly reinforced Pólya urns on a countable graph of uniformly bounded degree. The vertices have positive bounded firing rates, which need not be bounded away from zero. For every reinforcement exponent $\alpha\in[0,1)$ and every deterministic choice of finite positive initial weights, the normalized edge weights converge almost surely, coordinate-wise, to a deterministic vector independent of the initial weights. We identify this vector as the unique non-vanishing equilibrium of the limiting equation. This proves the homogenization conjecture of Couzinié and Hirsch, including its formulation for non-uniform firing rates in the MATRIX open-problems report. The main step is a Liouville theorem for the limiting equation, established by a logarithmic-norm comparison of complete solutions using the row-sum balance at each vertex.

The study of the distribution of real sequences in the unit torus has a long and rich history. There are many notions which describe the “pseudo-random” behavior of a given deterministic sequence (often of arithmetic origin), such as equidistribution (which describes the distribution on a “global scale”) or the pair correlation (which is a “local-scale” statistics). Another popular and important statistics, which can be applied to all ranges from global via intermediate to local, is the number variance $V_N(s)$, which describes the mean-square fluctuation of the number of elements of an initial segment of the sequence in an interval of length $s$, when moving this interval around the torus. In the terminology of quantum chaology, where such statistics are used to describe the distribution of the energy levels of a system, pseudo-random behavior is called Poissonian behavior. Since the number variance is an important indicator to qualify pseudo-random behavior, it is quite surprising that the literature does not seem to contain a precise description of the distribution and of the almost sure asymptotic behavior of the number variance in the truly random case (i.i.d.\ random points), even in the comparatively simple setup of uniformly distributed points on the unit torus. In the present paper we calculate the distribution of the number variance with high precision throughout a wide range of the parameter $s$, and determine the precise threshold (in terms of $s$) where the “Poissonian” asymptotics $V_N(s) \sim N s$ breaks for a typical realization of a random sequence. Key technical input comes from the Komlós–Major–Tusnády theorem and the Karhunen-Loève decomposition.