Papers

In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the $p$-logarithmic Sobolev constant vanishes at $p=1/2$ and a correlated graph-based example for which the $p$-logarithmic Sobolev constant tends to $-\infty$ as $p\to 0^+$. These examples obstruct a positive uniform reverse-hypercontractivity rate on $(0,1)$. By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for $1/2< p\leq q < 1$ with explicit sufficient times. For each fixed $\delta>0$, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on $[\delta, 1]$ and hence quantum reverse hypercontractivity for $\delta< p\leq q<1$ with fixed $0<\delta<1/2$. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for $\delta< p\leq q< 1$ with fixed $0<\delta<1$ for connected graphs when $n \geq 3$.

Adaptive arm selection changes the distribution of the observations collected by a bandit algorithm, but it need not change their limiting empirical spectrum. We study Gaussian bandit designs in which the dimension and the number of observations grow proportionally. A quantitative coupling theorem compares the design generated by any causal selection rule with an independent Gaussian design. If the logarithm of the number of available arms is sublinear in the dimension, the empirical spectral distribution converges to the Marchenko-Pastur law, uniformly over the selection rule. Consequently, Gaussian Bayesian bandits have policy-independent first-order limits for posterior mean-square uncertainty, squared posterior covariance, and information acquisition. For linear-score selection, we obtain the exact conditional arm distribution and show that two-arm selection produces an exactly Wishart Gram matrix in every dimension, despite its nonzero conditional mean. For a fixed selection direction, we identify an explicit eigenvalue and eigenvector transition governed by the second moment of a Gaussian maximum. A counterexample shows that a direction's overlap with a reference signal does not determine this transition. Finally, an exponentially large arm pool permits a different bulk limit, establishing the order-sharpness of the arm-growth condition. These results distinguish global spectral stability from directional effects in adaptive bandit data.

Let $(c_t)_{t\geq 0}$ be a deterministic family of edge conductances on a countable vertex set, monotone in $t$, and let $(X_t)$ be the random walk that takes its $t$-th step using the conductances $c_t$. We prove that if $c_t\uparrow c_\infty$ and $c_\infty$ is recurrent (respectively, $c_0$ is transient), then $(X_t)$ is almost surely recurrent (respectively, transient), i.e., visits every vertex infinitely (respectively, finitely) often. We also establish the analogous results in continuous time. This proves conjectures of Amir, Benjamini, Gurel-Gurevich, and Kozma, and the special case of $\{ 0, 1 \}$-valued conductances corresponds to simple random walk on a growing graph, and in this special case our results prove a conjecture of Dembo, Huang, and Sidoravicius. In addition, we provide counterexamples to the corresponding conjectures when $(c_t)$ is monotone non-increasing: if $c_t\downarrow c_\infty$ and $c_\infty$ is transient, $(X_t)$ need not be transient, and similarly if $c_0$ is recurrent, $(X_t)$ need not be recurrent, even if $c_\infty\geq\alpha c_0$ for some $\alpha>0$.

We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman-Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.

We initiate the study of approximating the top eigenvalue and eigenvector of a random symmetric matrix $ A \in \mathbb{R}^{n\times n} $ using $ q(A)b $ where $q$ is a degree-$d$ polynomial and $b$ is a standard Gaussian vector independent of $A$. For spiked GOE $ Y = \lambda vv^\top + X $, we identify $ d_\star = \frac{\log(n)}{2\log(\lambda)} $ to be the critical degree threshold above which accurate approximation of the top eigenvalue and eigenvector is possible. This sharpens the common belief that spectral methods can be implemented by $ O(\log(n)) $-step power iterations and offers a precise connection between spectral methods and low-degree polynomial algorithms, a popular proxy for all polynomial-time algorithms. For GOE $X$, we identify $ d_\star = n^{1/3+o(1)} $ to be the critical degree threshold for top eigenvector approximation, whereas constant degree suffices for top eigenvalue approximation. Moreover, in the limit where $ d/n^{1/3} $ converges to a positive finite constant, we compute the exact asymptotic eigenvector approximation accuracy in terms of the expected squared overlap. These results significantly improve upon predictions made in randomized numerical linear algebra for deterministic data matrices that the iteration count of power methods with random initialization is governed by the inverse spectral gap. Technically, our analyses leverage extremal properties of Chebyshev polynomials and draw upon the rich literature of random matrix theory.

We discuss in detail Neutral Measures, which are measures which believably generate any real in Cantor space. We intuitively build up the classical notions of randomness, and show that neutral measures do exist, using the language of continuous semimeasures. Finally, we construct some specific randomness tests which allow us to enforce de- sirable properties on neutral measures. As an application, we show the existence of Gibbs measures for a large class of finite range Hamilto- nians on lattice models, and establish their relationship with neutral measures.

We study the two-parameter exponential family associated with the complete-monotonicity problem of Alzer and Berg. Strict necessary parameter bounds place every possible Bernstein function in the domain of a regularized Laplace representation. A quantitative positivity-transfer inequality then yields a linear sufficient condition with the largest possible universal coefficient, expressed in terms of the exact one-parameter critical exponent. To describe the whole admissible region, we derive convolution identities for parameter derivatives and prove that increasing either normalized parameter destroys positivity at every zero of a nonnegative density. Combined with uniform tail estimates, this excludes gaps in the admissible parameter intervals and gives a continuous, strictly monotone optimal boundary. Global nonnegativity of the density and contact with zero characterize that boundary; its inverse determines the complete admissible interval for the second parameter. The characterization is implicit and requires neither uniqueness nor nondegeneracy of the contact points. Published numerical approximations of the one-parameter exponent are distinguished from the exact results and from the finite rational certificate used in the proofs. We also derive a variational formula and a rigorous framework for validated numerical enclosure of the boundary, and formulate a boundary-contact conjecture asserting uniqueness and quadratic contact at every interior boundary point.

We study global convergence guarantees of third-order Langevin dynamics for non-convex optimization via simulated annealing with fixed friction and decreasing noise. An explicit three-block distorted entropy transfers dissipation from the noisy auxiliary variable to the full state. Under dissipativity, regularity, and low-temperature functional-inequality assumptions, logarithmic cooling drives the objective values to the global minimum in probability at the barrier-controlled kinetic rate. For the exact-force-integral and midpoint three-stage discretizations, polynomially decreasing steps preserve this rate on the physical time scale. The cubic local endpoint estimate gives a less restrictive sufficient step-size condition than the available frozen-force kinetic result. A comparison with the one-gradient UBU integrator shows how its centered stochastic local error leads, under the same strong-coupling analysis, to a smaller sufficient iteration exponent. Numerical experiments are conducted to illustrate our theory. For a double well objective, third-order Langevin terminal-success point estimates are higher than UBU at both a common horizon and an equal gradient budget. For a high-dimensional nonconvex neural-network objective using synthetic data, independently tuned UBU and third-order Langevin schemes both outperform overdamped Langevin dynamics; the third-order Langevin point estimate is higher. For the same neural-network objective on real data, we show the same point-estimate ordering for best-basin probability and post-quench test accuracy. Numerical code and associated experiment results are publicly available at https://github.com/gagawjbytw/simulated-annealing-third-order-langevin.

Lecture notes from a graduate course given by Michael Krivelevich at the School of Mathematical Sciences of Tel Aviv University in the spring semester of 2026. Topics covered include: phase transition and the giant component in $G(n,p)$; long paths and cycles in supercritical and sparse random graphs; thresholds for connectedness and perfect matching; general model of a random subgraph of a finite graph; phase transition and the giant component in the random hypercube; polynomial diameter of the giant component; perfect matchings in the random hypercube.

We extend the Besov rough path framework of Friz and Seeger to paths lying in Besov spaces that are Muckenhoupt-weighted in time. We establish a weighted Besov sewing lemma, construct Young and rough integrals, and prove local Lipschitz continuity of the Itô-Lyons map in weighted Besov norms. The resulting estimates deliver "location-sensitive" control of errors in RDE solution increments. We illustrate the framework with a diffusion whose coefficient is singular at a specified time.

The Baringhaus-Henze-Epps-Pulley (BHEP) tests for multivariate normality are affine-invariant goodness-of-fit tests based on a Gaussian-weighted $L^2$ distance between empirical and Gaussian characteristic functions. In 1990, Henze and Zirkler expressed the limiting null distribution through the eigenvalues of an integral operator on the standard Gaussian space. In 1997, Henze and Wagner obtained a simpler covariance kernel and raised the problem of calculating the eigenvalues of the resulting operator on a Gaussian-weighted space. Although subsequent work treated the univariate case and numerical approximations in a few low dimensions, the complete all-dimensional spectral problem remained open. This paper determines both complete spectra for every dimension $d \in \mathbb{N}$ and every smoothing parameter $\beta > 0$. The two operators are shown to have the forms $\mathcal{X}_{\beta,d}^*\mathcal{X}_{\beta,d}$ and $\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^*$ for the same Hilbert-Schmidt operator $\mathcal{X}_{\beta,d}$. Consequently, their nonzero eigenvalues agree, including multiplicities, while the null space of the Henze-Zirkler operator is identified exactly. The Gaussian integral operator in the Henze-Wagner decomposition is diagonalized by Mehler's formula, and rotational symmetry confines the finite-rank correction to the sectors associated with spherical harmonics of degrees $0$, $1$, and $2$. The degree-$1$ and degree-$2$ eigenvalues are characterized by scalar transcendental equations, and the radial eigenvalues by an explicit pole-safe Fredholm determinant. The paper establishes nonnegativity, multiplicities, eigenfunction reconstruction, completeness, the trace identity, and a complete characterization of all exceptional pole cases.

Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for every $t\in\mathbb{R}$, and for each fixed $t>0$ equality holds only when $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance $1/(n-1)$, which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider $n$ equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least $n-1$. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.

Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.

We prove results about stability under mixtures for transport–entropy inequalities and restricted functional inequalities of Poincaré and log-Sobolev type. We treat Talagrand inequalities of type $\mathbf{T}_2$ and $\mathbf{T}_1$ together with their generalizations, as well as weak transportation inequalities introduced by Gozlan et al. Our main results provide mild sufficient conditions under which a given inequality may be transferred from mixture components to the mixture itself.

We establish the natural scale of instability in real Gaussian phase retrieval at the critical injectivity threshold. Let $A$ be a $(2M-1)\times M$ matrix with independent standard Gaussian entries and let $K_M=\binom{2M-1}{M}$. For every $w_M\to\infty$ with $\log w_M=o(M)$, we prove that $\mathbb{P}\{(w_M\sqrt{M}K_M)^{-1}\le\omega(A)\le w_M/(\sqrt{M}K_M)\}\to1$, where $\omega(A)$ is the Balan–Wang stability parameter. Consequently, $-M^{-1}\log\omega(A)\to\log4$ in probability. For full-spark matrices at this threshold, $\omega(A)$ equals both the optimal lower Lipschitz constant of $x\mapsto|Ax|$ and the minimum least singular value over all square row submatrices. The upper bound follows from a second-moment analysis of overlapping minors. A weighted Gaussian inverse-tail asymptotic and inverse-Wishart concentration yield asymptotic independence for central overlaps; rectangular hard-edge bounds control the remaining overlaps. The matching lower bound follows from a union bound and a square Gaussian hard-edge estimate.

A class of models which does not require additional operators on the language of classical sentential calculi to capture some notion of necessity, possibility, and impossibility is introduced and investigated.

For the sinh-Gordon model (without external mass term), we show that the log-Sobolev inequality holds uniformly in the lattice and volume regularisations for all parameters in the regime in which the Gaussian multiplicative chaos has second moments. This implies that infinite-volume “massless” sinh-Gordon measures on $\mathbb{R}^2$ satisfying all Osterwalder–Schrader axioms exist and that they have exponential decay of correlations (are “massive”). The proof uses the Polchinski equation method for log-Sobolev inequalities. The required estimates on the renormalised potential are established using simple perturbation theoretic bounds, valid for absolutely monotone potentials, and an extension of the correlation inequality of Ding–Song–Sun to potentials in the GHS class. This inequality is proved also by using a variant of the Polchinski equation through the maximum principle.

We establish a Donsker-Prokhorov invariance principle in some Besov spaces. Specifically, we show that polygonal line processes associated with partial sum processes of a triangular array of row-wise independent random variables converge in distribution to Brownian motion, extending earlier results in the literature.

We study the convergence rate of the heat-bath Glauber dynamics for the Discrete Gaussian Free Field on arbitrary connected finite graphs. We show that, when starting from super-harmonic initial conditions, the evolution enjoys a strong form of monotonicity. This allows us to get a sharp mixing profile as the size of the graphs diverges. More precisely, we show that mixing occurs at time $\frac{1}{2\lambda}\log(\mathcal{E})$ with window $\mathcal{O}(1/\lambda)$, where $\lambda$ is the spectral gap of the graph Laplacian and $\mathcal{E}$ is the energy of the super-harmonic initial condition. This result holds for arbitrary graphs that do not exhibit extreme connectivity properties (one way or the other). In particular, it holds for finite boxes of the grid $\mathbb{Z}^d$, in dimension $d\geq 3$.

We prove an upper bound on the homogenization length scale which is a fixed power of the coarse ellipticity ratio for coefficient fields that satisfy a finite range of dependence assumption.

We study a discrete-time auction model in which multiple sellers update their bids according to their performance in the preceding round. Bidders aim to maximize their profits and adjust their bids solely based on their most recent outcome (myopic behavior). In each round, the auctioneer purchases the lowest-priced $p$-fraction of the total quantity offered by the bidders. We find a system of differential equations governing the macroscopic dynamics and derive it as a scaling limit of the microscopic model. We find an explicit solution for the max-price evolution $q_t$ and show that, in the long run, bidders coordinate, i.e., their bids converge to a common value depending only on their initial distribution and the fraction $p$. For Poisson-distributed initial bids, we establish hydrodynamic limits for the empirical bid distribution and the max-price trajectory and conjecture the corresponding Gaussian fluctuations for $q_t$. Finally, we generalize the model to allow for heterogeneous bid-update velocities: in this case, the max-price velocity becomes proportional to the harmonic mean of the update velocities of bidders at the max-price.

Blest's rank correlation $\nu$ is a variant of Spearman's rho $\rho$ that weights the leading ranks of one variable more heavily, at the price that $\nu$ is not symmetric in its arguments. We quantify both features by determining the exact region of $(\rho,\nu)$ over all bivariate copulas, as well as that of $(\eta,\nu)$, where $\eta$ is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs $(\nu(C),\nu(C^\top))$ formed by a copula $C$ and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most $1/4$, and interchanging the two variables changes it by at most $27/64$, improving on the bound $1/2$ implied by the first inequality. For every given value of $\rho$ or $\eta$, each corresponding extreme value of $\nu$ is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the $(\eta,\nu)$-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.

A coherent radar observes sea clutter through a non-injective multiplicative combination of latent texture and coherence variables. We study a class of partially observed slow–fast diffusions for which the projected quadratic covariance has the form \(Q=S+qyy^\top\), using Field's sea-clutter diffusion as a solvable benchmark. For fixed-shape fast covariance, we derive an exact generalized spectral-gap criterion for recovery of the latent slow coordinate and a stability bound. When the fast covariance changes shape, we obtain an invariant scale–shape decomposition and exact conditions for reinforcement or cancellation of the latent quadratic signature. We then quantify the loss of resolvability caused by finite sampling and receiver noise. Controlled CIR–OU simulations verify the analytic identities and exhibit regimes in which structural observability does not yield stable reconstruction.

Score-based diffusion models are increasingly considered in settings where the underlying data distribution may differ from the training distribution, yet existing theoretical guarantees largely focus on the no-shift setting. In this work, we study robust score estimation under Wasserstein perturbations of a reference distribution. For the Ornstein–Uhlenbeck diffusion, we show that robust estimation decomposes into two fundamental components: the statistical cost of learning the reference distribution and the intrinsic cost of distribution shift. The latter scales quadratically with the Wasserstein radius, and this dependence is minimax optimal. We construct an explicit finite-sample estimator achieving the resulting robust minimax rate without knowing the shift radius. When the reference distribution lies on an unknown low-dimensional subspace, the statistical term adapts to the intrinsic dimension while the shift cost remains unchanged. Finally, we show that the same decomposition governs positive-time reverse sampling and obtain matching minimax guarantees in KL divergence. Together, these results characterize how finite data, intrinsic dimension, and distribution shift affect the robustness of score-based diffusion models.

Algorithms in multi-agent systems such as federated learning, mobile robotic swarming, and consensus control can be designed and analyzed as distributed stochastic approximation algorithms. Such algorithms involve information exchanges between agents for various computations. The freshness of the information can be quantified using the Age of Information (AoI) metric. Consider robotic teams operating in highly obstructed geographical settings, such as subterranean or dense urban environments. Because of spatial disconnections, AoI has empirically been observed to be heavy-tailed with unbounded moments. However, most analyses assume AoI with bounded moments, creating a gap between theory and practice. To the best of our knowledge, ours is the first analysis under general heavy-tailed AoI with potentially infinite mean. We study the stability (almost sure boundedness of the distributed iterates) and convergence of multi-agent systems that are strictly dissipative in the scaling limit (system at “infinity”). Examples include most gradient-based and consensus algorithms under the Robbins-Monro step-size regime.

We study strong measurability and adapted approximations for maps taking values in possibly nonseparable metric spaces. After recalling the metric-space Pettis criterion, we give conditions under which measurability implies strong measurability. Under the continuum hypothesis (CH), every measurable map from a countably generated measurable space into a metric space is strongly measurable for every measure on the domain. For finite or $\sigma$-finite measures, the same conclusion holds when the target density is strictly smaller than every real-valued measurable cardinal. Under CH this includes targets of density at most $\mathfrak c$; if no real-valued measurable cardinal exists, it holds for all metric targets. For function-valued maps, we show that pointwise measurability and continuity of the sections need not ensure measurability in the function-space topology. We provide a verification criterion using countably many evaluations and essential separability of the range. Further, we obtain elementary $L^p$-approximations of progressive or jointly measurable adapted processes under suitable filtration assumptions, Lipschitz functionals with path and Wasserstein variables, and smooth approximations based on finitely many observations of a process with left- or right-continuous paths. Finally, under CH and suitable assumptions, we apply these results to controlled stochastic differential equations in separable Hilbert spaces, allowing for jumps, random coefficients, path dependence, and state- and control-law dependence; we establish convergence of states and costs uniformly over admissible controls, convergence of optimal values, and transfer of near-optimal controls.

Let $G$ be the line graph of a finite simple graph, with $n\geq1$ vertices and maximum degree $\Delta$. We prove that single-site Glauber dynamics for uniform proper $q$-colorings mixes in $O_\Delta(n\log(n/\varepsilon))$ steps for every integer $q\geq\Delta+5$. Our proof uses the Bochner framework of Chen and Liu (2026).

Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n$, and let $S=\sum_i B_i$. Hillion and Johnson proved that the Shannon entropy of $S$ is jointly concave in the parameter vector and proposed corresponding critical-order conjectures for R'enyi and Tsallis entropies, with predicted thresholds $2$ and approximately $3.65986$, respectively. We determine both thresholds exactly. For every $0<q<1$, the power sum $\sum_k \mathbb P(S=k)^q$ is jointly concave in $(p_1,\ldots,p_n)$, and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order $q$ are jointly concave. At $q=1$ this agrees with the Shannon theorem. For every $q>1$, joint concavity fails already for the sum of two Bernoulli variables: a transverse interpolation in which the two parameters move in opposite directions gives strict local convexity for both entropies. Hence the universal joint-concavity range for both families is exactly $0<q\leq 1$. Below order one, the proof combines the Hillion–Johnson transport inequality with an explicit nonlinear telescoping correction. The corrected local curvature reduces to a two-dimensional quadratic form. An exact Riccati identity, together with a one-sided zero-crossing argument, proves positivity of its determinant throughout the full range $0<q<1$.

A latent severity has a bounded lower tail with density of shape alpha and scale L. It is observed only through a fixed Markov kernel K that is biased and non-additive. The relative conditional spread of K diverges at the endpoint. Our sample is i.i.d. from the marginal Q alone, with no anchoring covariate or instrument. We prove a dichotomy. The shape index alpha is identifiable: for every admissible choice of the class constants, any two observationally equivalent members of a lean class share alpha, determined by a near-endpoint expansion of Q. The rate, namely L and the fixed-scale exceedance p_tau, does not survive. There exist admissible shared class constants and two members of a smaller regularity class whose observed laws coincide exactly. Across the pair alpha agrees, whereas L and p_tau move. A degenerate Le Cam two-point bound excludes any uniformly consistent estimator of either, and pointwise consistency fails at one member. Only the rate needs an anchor. We conjecture that a known kernel family with known edge map identifies the rate fiber by fiber if and only if the family satisfies a fixed-scale injectivity clause, and we prove the sufficiency direction. In surrogate safety, uncalibrated conflict data give the shape of near-crash risk, not its absolute rate.

Let $P(t)$ be the transition function of an absorbing continuous-time Markov chain on a countable state space. We study asymptotic relations of the form $p_{ij}(t)\sim a_{ij} L(t)$, $t\to\infty$, where $L$ is independent of $i$ and $j$. We obtain general conditions under which the coefficient matrix $A=(a_{ij})$ has rank one and describe the resulting consequences for survival probabilities and conditional distributions. The analysis applies to both reducible and irreducible chains. We also construct an irreducible counterexample, based on a killed random walk on a homogeneous tree, for which a common asymptotic scale exists but the coefficient matrix has rank greater than one. This shows that irreducibility alone does not imply rank-one asymptotics. The results identify conditions under which fixed-state transition asymptotics determine the asymptotic behaviour of the entire transition function and clarify the limitations of such conclusions in the absence of additional structure.