Papers

In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent $\alpha>0$. The covariance of the Gaussian environment $\{\omega_n\}_{n\in\mathbb N}$ is given by $\text{Cov}_{\mathbb P}(\omega_n,\omega_m)\sim |n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $\alpha\in(0,\frac12]$, $H\in(\frac12,1)$ and $\alpha+2H>2$, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the $L^1$-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not $L^2$-integrable when $\alpha<\frac12$.

We study the large-depth behavior of residual networks whose weights are correlated across layers at initialization. Our results confirm and extend a conjecture of Marion et al. [2025], according to which correlated initializations should interpolate continuously between the Brownian stochastic differential equation arising from independent initialization and the ordinary differential equation arising from perfectly correlated initialization. When the initialization is obtained from the application of a feature function to a stationary Gaussian sequence with regularly varying correlation, we prove that there exists a unique critical scaling such that the infinite-depth limit is the solution of a Young differential equation driven by a Hermite process. Hermite processes reduce to the fractional Brownian motion if the feature function generating the initialization has Hermite rank one, which is the case for the identity function, for example. We show that the critical scaling and asymptotic limit are uniquely determined by the decay of correlations together with the Hermite rank of the feature function. Consequently, the correlation structure and Hermite rank of the initialization represent meaningful hyperparameters in the asymptotic regime. By contrast, under finite-variance iid initialization, the asymptotic driver is universally Brownian up to normalization regardless of the choice of distribution. Our proofs rely on a collection of novel results establishing a robust stability theory for Young differential equations in Banach spaces.

We consider the Schrödinger equation for a charged particle in a randomly fluctuating external magnetic field and establish a singular perturbation limit in which the solution converges to a deterministic Schrödinger equation with Gaussian fluctuations. We propose a stochastic Schrödinger equation incorporating the behavior of both the deterministic average and the fluctuations and show that it has the same limiting behavior as the original model, while providing a simpler way to study the distribution of relevant observables.

For every $\beta>0$, we establish multi-time soft-edge moment convergence for the Jack–Plancherel process, a discrete $\beta$-analogue of Dyson Brownian motion. The limiting moments admit an absolutely convergent expansion in terms of nonnegative Brownian bridges decorated by pairs of equal and opposite jumps. At equal times, we identify this limit with the joint Laplace statistics of the Airy$_\beta$ process for $\beta\geq1$. The same Brownian expansion yields an asymptotic $\beta$-topological expansion of the marginal $b$-conjecture and, at $\beta=2$, an explicit nonnegative formula for the Witten–Kontsevich intersection numbers.

We introduce a conjecture for Bernoulli bond percolation on finite graphs. Roughly speaking, it asserts that if each boundary vertex is associated with a highly probable event that is increasing with respect to the percolation configuration, then, conditionally on the origin being connected to the boundary, the origin is likely to be connected to a boundary vertex whose associated event occurs. The conjecture implies a high-probability connectivity statement which is known to imply $\theta(p_c)=0$ for Bernoulli bond percolation on $\mathbb Z^d$, for every $d\geq2$. We prove the conjecture for finite planar graphs when the origin and the boundary vertices lie on the outer-face boundary, with the explicit bound $1-2\sqrt{\varepsilon}$. The proof combines a left-first depth-first exploration with a partial FKG inequality adapted to monotonicity up to a stopping time. A counterexample shows that the connectivity structure is essential: the analogous statement fails when the connectivity events are replaced by arbitrary increasing events.

Consider $k$ independent Bernoulli populations, each sampled $n$ times, and select the $t$ populations with the largest success counts, breaking ties uniformly. Classical monotonicity reduces the worst case over the preference zone with separation $\delta$ to the slippage family with levels $p$ and $p+\delta$, leaving only its absolute location $p\in[0,1-\delta]$ undetermined. A Gaussian approximation suggests the symmetric center $p_{\mathrm c}=(1-\delta)/2$, and the exact two-population problem is uniquely centered there for every $n\ge2$. For fixed $k,t$ and $\delta\in(0,1)$, we prove that exact eventual centering holds precisely when $k=2t$. When $k\ne2t$, the least-favorable location $p_{n,k,t}^*$ satisfies \[ p_{n,k,t}^*-p_{\mathrm c} =(k-2t)C_\delta n^{-1/2}e^{-n\Gamma_\delta}\{1+o(1)\}, \] where $C_\delta$ and $\Gamma_\delta$ are explicit and positive. In either case, the least-favorable location is eventually unique. The proof writes incorrect selection as a union of pairwise misrankings and applies inclusion–exclusion, yielding a bipartite graph expansion. A single misranking has its exact maximum at the symmetric center and determines the central curvature; two-edge intersections sharing one population determine the central slope through their multiplicity imbalance; all remaining graphs have higher large-deviation rates.

We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.

The contact process with avoidance is a generalization of the classical contact process (SIS epidemic) that introduces a mechanism for healthy individuals to avoid their infected neighbors. Let $G$ be a directed graph. At each time $t$, each vertex is either healthy or infected and each edge is either active or inactive. Each infected vertex infects each healthy neighbor across each active edge at rate $\lambda$ and recovers at rate $1$, while each active edge pointing from an infected vertex to a healthy vertex becomes inactive at rate $\alpha$. An inactive edge becomes active when its tail vertex recovers. This model has been previously studied on $\mathbb{Z}$, the $n$-cycle $\mathbb{Z}_n$, and the $n$-star graph; here we extend the study of this model to lattices $\mathbb{Z}^d$, $d \geq 2$. We show that for every $d \geq 2$ and fixed $\alpha > 0$, there exist constants $\lambda(\alpha,d)^-$ and $\lambda(\alpha,d)^+$ such that for all $\lambda < \lambda(\alpha,d)^-$ the infection dies out almost surely while for all $\lambda > \lambda(\alpha,d)^+$ the infection persists indefinitely with positive probability. Furthermore, we show that both $\lambda(\alpha,d)^-$ and $\lambda(\alpha,d)^+$ scale like $1/d$ as $d \rightarrow \infty$ and that there exists a constant $C(\alpha)$ such that when $\lambda > C(\alpha)/d$ the process has a nontrivial invariant measure for $d$ sufficiently large. Our methods and most of our results also apply to the SIRS model and a related model in which infected vertices enter an isolated state at rate $\alpha$ and transition from both isolated and infected to healthy at rate $1$.

Let $D=2^n$ and $M=3n+9\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\to0$ and $MQ_D^{3/4}/D^2\to0$, then, for every fixed $0<x<\pi/\sqrt3$, \[ \mu_D\bigl(B_{Q_D}([I],x\sqrt{Q_D})\bigr)\le e^{-c_xD^2}. \] Here $\mu_D$ is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to $(\pi/\sqrt3)\sqrt{Q_D}$ throughout the window $M\ll Q_D\ll D^{8/3}M^{-4/3}$. Choosing $Q_D=\kappa D^{8/3}M^{-4/3}$ with sufficiently small fixed $\kappa>0$ yields a Haar-typical lower bound of order $D^{4/3}M^{-2/3}$ for the corresponding infinite-cliff Carnot–Carathéodory distance, outside an $e^{-\Omega(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4/3$, disproving Brown's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.

We study the problem of estimation of the parameter in a stochastic wave equation driven by an infinite dimensional fractional Brownian motion.

We study a mean-field optimal stopping problem with an endogenous population-level shutdown. All remaining agents stop when the survival mass falls below a prescribed threshold. We recast the discontinuous objective as the singular, nonconvex constraint that the survival mass lie in $\{0\}\cup[\alpha,1]$. We prove the equivalence of strong and weak values via an approximation and the existence of an optimal rule via compactness and penalization. We also prove a dynamic programming principle. The value is continuous away from the critical boundary but may be discontinuous at the boundary itself. Under strict initial feasibility, finite-population values converge to the mean-field value. In the same regime, the laws of near-optimal empirical measures are tight and every mean-field optimizer admits a recovery sequence. At the threshold, however, finite-population convergence may fail.

Within probability-coordinate geometry, a continuous distribution function restricted to an interval can serve as a probability coordinate chart. Averaging in these coordinates defines probability barycenters and associated Kolmogorov moments. Because the class of admissible charts is infinite dimensional, this paper develops a finite-dimensional approximation theory. Relative to a fixed base chart, every chart on the same interval is represented uniquely by a strictly increasing deformation of the unit interval. Monotone piecewise-linear deformations are shown to be exact convex combinations of elementary ramp functions, thereby parametrising probability geometries by allocation vectors in a simplex. Such geometries are dense in the admissible class. We study two constructive schemes: ramp interpolation and Bernstein approximation. Bernstein approximants also admit a ramp-dictionary representation with smooth beta atoms derived from uniform order statistics, while automatically preserving admissibility. Uniform approximation of deformation maps yields explicit stability bounds for probability barycenters and initial Kolmogorov moments. For deformed intrinsic charts, the barycenter is a quantile whose level depends only on the deformation, connecting the construction with chart-based quantile estimation. Deterministic numerical illustrations demonstrate both approximation schemes and their induced barycenter errors.

Let X be a centered spectrally positive infinitely divisible random variable with finite nonzero variance V. We determine the smallest scale c in a right-sided sub-gamma bound whose quadratic proxy is fixed at V. A probability transform of the normalized Kolmogorov canonical measure produces a nonnegative variable R_X for which log E e^{tX} = V t^2 M_{R_X}(t)/2. The scale problem therefore reduces to comparing the MGF of R_X with that of an exponential law, leading to an exact one-dimensional variational formula. If X is non-Gaussian and has a positive exponential moment, the optimal scale is at least kappa_3(X)/(3V), with equality precisely when R_X belongs to the M-class of Klar and Mueller. Equality in the standard sub-gamma envelope occurs only for the Gaussian law or, in the non-Gaussian case, for centered compound Poisson laws with exponential jumps. For a general compound Poisson law, R_X is the second-order equilibrium distribution of the jump size. Exact calculations for gamma and bounded two-point jumps show that the optimum may occur at the origin, at the positive MGF boundary, or at an interior point.

We consider a weak martingale optimal transport problem related to the martingale analogue of the Benamou–Brenier formula. In contrast to the classical setting, the second marginal is not given, but specified only through a finite number of moment constraints of a particular form. For this problem, we derive a finite-dimensional dual formulation and prove the absence of a duality gap under the specified interiority condition on the constraint vector. In addition, we describe the structure of the optimal martingale coupling and its relation to Bass martingales.

We show that the $E_8$ and Leech lattices generate the stealthiest isometry-invariant, locally square-integrable point processes of intensity $1$ in $\mathbb{R}^8$ and $\mathbb{R}^{24}$, respectively, and that they are the unique stealthiest processes. We also apply the proof technique to sphere packing and energy minimization, and we construct a $20$-dimensional periodic configuration that is stealthier than any known $20$-dimensional lattice with the same particle density. This periodic configuration is a formal dual of Vardy's $20$-dimensional sphere packing.

Although market participants generally have access to a common information set, they make decisions based on forecasts formed over heterogeneous horizons. Because market impact depends on aggregate positions rather than trader identities, these decisions feed back into prices through their collective effect. We introduce a linear mean-field model of this interaction. The observed price is decomposed into a martingale component, a common predictable signal represented by a Volterra process, and the market impact generated by aggregate positions. Agents take positions according to conditional forecasts of future signal increments and a fraction of anticipated aggregate impact over their respective horizons. Within a Gaussian-Volterra framework, we characterize equilibrium through a linear fixed-point equation for aggregate positions and establish existence and uniqueness under explicit conditions. At equilibrium, we identify a balance condition that cancels the direct transmission of the common signal to the observed price. We then study the limit in which agents fully account for market impact. Along a suitably scaled family of equilibria satisfying explicit conditions, the contributions of the predictable signal and the market impact cancel in the limit, and the observed price converges to its martingale component. For fractional-type signals and Gamma-distributed horizons, we further derive local Hölder bounds and identify the horizon distributions for which the observed price has local regularity compatible with that of Brownian motion.

Data-consistent inversion (DCI) constructs probability measures whose push-forward distributions agree with observed data, while iterative data-consistent inversion (iDCI) extends this framework to generalized stochastic inverse problems by enforcing multiple push-forward constraints sequentially. Although iDCI avoids the direct approximation of high-dimensional joint densities, its relationship to the original joint DCI solution has remained unclear. In this work, we establish this relationship through copula theory. Using Sklar's theorem, we derive a factorization of the DCI update into separate marginal and dependence transformations and show that the discrepancy remaining after convergence of the iDCI algorithm is entirely characterized by the copulas associated with the observed and predicted joint distributions. This characterization motivates a copula-transformed iDCI solution, and we prove that an exact copula transformation recovers the original DCI solution. We further establish convergence results for approximate copula transformations under converging sequences of reference measures and progressively enriched feasible sets. Numerical examples demonstrate how the geometry induced by the quantity-of-interest map governs the importance of the copula transformation, illustrate an adaptive reference-measure refinement strategy for improving computational accuracy under a fixed sampling budget, and demonstrate the progressive refinement of generalized stochastic inverse problems through heterogeneous, asynchronously acquired experiments.

The divisible sandpile on $\mathbb{Z}^d$ starts from i.i.d. masses at each site, and, in each discrete time step, a site with mass above one keeps one unit and sends the excess equally to its neighbors. Levine, Murugan, Peres and Ugurcan (2016) showed that at mean one this process explodes, with every site emitting infinite mass. We show that the mass emitted from a site by time $t$ is of order $t^{(4-d)/4}$ for $d\leq3$, of order $\log t$ for $d=4$, and a tail-dependent, divergent rate for $d\geq5$. We further show that the mass emitted, after diffusive rescaling, converges to a Brownian optimal-stopping value for $d\leq3$ and to tail-dependent, weighted membrane fields for $d\geq5$, while at the critical dimension $d=4$, after superdiffusive rescaling, it converges to the membrane model. Using these estimates, we prove that, for every $d\geq2$, the set of sites that topple contains an infinite component at some mean below one, hence it has a non-trivial percolation phase transition. This answers a variant of a question of Fey, Meester and Redig (2009). The proof adapts ideas from the theory of level-set percolation of strongly correlated Gaussian fields.

In the parking model, each site of the $d$-dimensional lattice independently starts with one car with probability $p$ or one parking spot with probability $1-p$. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime $p=1/2$, the expected number of visits to a site in $n$ rounds is of order $n^{(4-d)/4}$ for $d\leq3$ and $\log n$ for $d\geq4$. We also prove that in the subcritical regime $p\in(0,1/2)$, the parking-time tail is bounded above and below by stretched exponentials with exponent $d/(d+2)$. As $p\uparrow1/2$, we also determine the divergence of the expected total number of visits to a site: its order is $(1-2p)^{-3}$, $(1-2p)^{-1}$ and $(1-2p)^{-1/3}$ in dimensions one, two and three, respectively, and $\log(1/(1-2p))$ in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of Levine and Peres plus a martingale-type term. These results answer questions posed by Damron, Gravner, Junge, Lyu and Sivakoff (2019).

One of the most common descriptions of solutions of singular SPDEs is their characterisation as the limit of solutions of renormalised smooth random PDEs. We quantify the speed of this convergence in the case of the KPZ equation. In particular, we show that the naïve guess that the rate is given by the distance of the noise regularity from the endpoint regularity for well-posedness is not correct. Instead, we obtain convergence at the larger rate $1/2$ and show that this rate is sharp. This is achieved by considering the equation satisfied by the rescaled error, which is critical for variance blowup, and showing that this equation has a limit given by an affine linear singular SPDE driven by a new, independent noise. This result can be alternatively interpreted as identifying the asymptotic size and law of fluctuations of the solutions of the KPZ equation driven by mollified noise around their singular limit.

We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert–Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman–Fredholm determinant of the centered interaction operator. We show that the Hilbert–Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert–Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.

We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.

The study of evolution in temporally fluctuating environments often relies on the analysis of Lyapunov exponents, which quantify the exponential growth of populations. However, when model parameters depend on a stochastic process, calculating the selection gradient, a key tool for predicting the evolution of phenotypic traits, becomes a mathematical challenge. While the periodic case has been resolved, a general approach for random environments remains to be developed. This article proposes a rigorous method to: Establish the differentiability of the leading Lyapunov exponent with respect to a parameter, providing an explicit integral formula for its derivative. Develop a numerical algorithm to approximate the derivative by solving an extended differential equation. Apply these results to the analysis of mutant invasion in a resident population at equilibrium, identifying the selection gradient as the derivative of the top Lyapunov exponent.

In this paper, we study the large deviation principle (LDP) for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the "bad" cut norm. Using the LDP, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. As applications, we focus on several concrete examples, which include monochromatic subgraph counts in sparse random graphs, Erdős-Rényi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.

We investigate the power-law decay of the survival probability of a Brownian particle diffusing in an infinite planar wedge whose two sides are partially reactive and described by Robin boundary conditions. We employ matched asymptotic analysis to relate the long-time asymptotics to a stationary harmonic Robin problem near the apex. This approach determines the persistence exponent for arbitrary opening angles and yields the prefactor explicitly for the family of wedges with $\alpha=\pi/n$ ($n=1,2,\ldots$). A simple extension of the apex prefactor to arbitrary angles is conjectured and supported numerically. This asymptotic behavior describes the crossover to the well-known result for perfectly absorbing wedges. As immediate applications, we also deduce the long-time behavior of the probability density function of the boundary local time on wedge sides, as well as the probability density function of the associated first-crossing time.

The pairwise independent correlation gap is the ratio of the maximum expected value of a set function under arbitrary dependence to that under pairwise independence, measuring the loss from this independence restriction. Under mutual independence, this gap is universally bounded by $e/(e-1)$ for monotone submodular functions. With pairwise independence, a tighter $4/3$ upper bound was established for several special cases, including $n=3$, and conjectured to hold universally. A recent AI-assisted counterexample disproved this conjecture for $n=5$, leaving the validity of the $n=4$ bound and the tight worst case bound open. We resolve both questions. First, for $n=4$, we establish that the $4/3$ bound holds universally and is tight using an AI-assisted proof combining theoretical analysis and computational verification. The proof combines a structural characterization of optimal numerator vertices, permutation symmetry, cone certificate systems, Bernstein polynomial representations, recursive simplex subdivision, and verification of $2,745$ Bernstein coefficient systems. Second, we show that the worst case pairwise independent correlation gap attains $e/(e-1)$ asymptotically by constructing an instance with identical marginal probabilities and a monotone submodular union coverage function on a ground set partitioned into $m$ blocks. The number of blocks grows sublinearly with the ground set size. The result follows by constructing a feasible solution to a scaled asymptotic reduced dual of the pairwise independent linear program and immediately extends to $t$-wise independent random elements ($t\ge2$), since $t$-wise independence implies pairwise independence. Thus, pairwise independence, despite being the least restrictive form of independence in the $t$-wise independence hierarchy, can be as restrictive as mutual independence in the worst case.

We study the classical $XY$-model on random geometric graphs $\mathcal{G}_{n, \varepsilon}$, which are obtained by sampling $n \in \mathbb{N}$ independent points in a finite domain $\Omega \subset \mathbb{R}^d$, $d \geq 2$, and connecting two points by and edge if their distance is of order $\varepsilon > 0$. We refer to $\mathcal{G}_{n, \varepsilon}$ as the random environment. Letting $\varepsilon \to 0$ as $n \to \infty$ at a sufficiently slow rate, these graphs capture the geometry of $\Omega$. Denoting the inverse temperature by $\beta$, we show that in the limit $\beta \to \infty$ at a rate depending on $n$ and $\varepsilon$, the $XY$-model on $\mathcal{G}_{n, \varepsilon}$ exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as $n \to \infty$. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp–Lieb inequality.

Tipping points—abrupt, potentially irreversible reorganizations of a system's statistical state—are commonly anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectra redden. This paradigm is powerful near simple equilibrium bifurcations but is not a general theory for stochastic, multistable, or metastable systems. We develop such a theory from the Ruelle–Pollicott (RP) spectrum of Kolmogorov generators of hypoelliptic Itô diffusions. Resolving the sensitivity of invariant statistics on RP spectral blocks shows that each contribution factorizes into a spectral denominator and a residue coupling the block to both the observable and perturbation direction. Small denominators permit large responses; residues produce them. Thus a closing RP gap is not sufficient for an early warning, while growing residues can generate the classical signature with no gap closure. An early-warning signal is therefore a property of a triple: RP block, observable, and perturbation direction. We demonstrate this on a stochastic non-normal system whose RP spectrum is exactly frozen, yet classical indicators become more alarming than during genuine gap closure. The RP decomposition attributes this to residue growth and yields an index $\mathcal{N}(f)$ satisfying $\mathcal{N}(f)\leq 1$ for reversible dynamics; hence $\mathcal{N}(f)>1$ certifies residue-driven amplification. For metastable systems, one killed problem yields two complementary spectral objects: the Doob $Q$-process isolates in-well recovery, while escape clocks and committor-weighted destination probabilities govern interwell transitions. In a one-dimensional fold, these scale as $(\epsilon_c-\epsilon)^{1/2}$ and $(\epsilon_c-\epsilon)^{3/2}$, separating bifurcation-induced from noise-induced tipping. Doob drift also connects to optimal Girsanov sampling.

Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound for <t,X> whose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2) multiplier give an exact variational formula for C_X. The resulting extended-valued function is positive homogeneous and is pointwise least among all homogeneous denominators compatible with the covariance quadratic form. We prove linear-map, convolution, and Levy-time rules, and show that the full family of directional remainders determines the law of X. Geometrically, C_X lies between the Minkowski functional of the moment-generating-function domain and one third of the positive support function of the Levy measure; the upper constant is sharp, and the zero set is a polar cone. The pole need not be subadditive. It is continuous on the sphere under global exponential moments and positive-definite covariance, whereas finite variance alone permits a jump from zero to infinity in nearby directions. For additive gamma-ray models, C_X equals the domain gauge and has a finite-polytope formula.

Fixing the quadratic proxy in a sub-gamma bound at the true variance leaves a scale to optimize, and a two-sided infinitely divisible law generally requires different scales in the two directions. For a centered law with finite nonzero variance, we normalize its Kolmogorov canonical measure and multiply the resulting variable by an independent Beta(1,2) variable. The signed remainder obtained in this way gives exact variational formulas for the right and left scales. We prove that a directional scale vanishes exactly when the Levy measure has no jumps in that direction, establish reflection, scaling, convolution, Levy-time, and opposite-jump perturbation rules, and recover the Levy triplet from the remainder law. The formulas give the two Gamma scales for bilateral Gamma laws and, for centered Skellam laws, the exact transition points p_+=(2+sqrt(3))/4 and p_-=(2-sqrt(3))/4 between local and interior control; when positive jumps are rare, the right scale is asymptotic to 1/log(1/p). The pair therefore records jump direction and the mechanism that controls the variance-exact sub-gamma pole.