Papers

Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is close in cut distance to the Erdős–Rényi random graph $G(n,p)$. We prove that this region is twice as large as the corresponding region for the unconditional model. To study refined properties of the conditional model, we then analyze the global Kawasaki algorithm for sampling from it. Within the replica symmetric region, and under the additional condition that $4\beta p(1-p)<0.5$ when $|p-1/2|\lesssim 0.4632$, where $\beta$ is the model parameter, we prove metastable fast mixing of the Kawasaki algorithm. As corollaries, we obtain a weak Poincaré inequality and use it to deduce concentration inequalities of optimal order for conditional subgraph counts. The additional condition $4\beta p(1-p)<1/2$ if $|p-1/2|\lesssim 0.4632$ comes from our proof technique of contractive coupling. This bottleneck did not appear in previous studies applying the contractive coupling technique to unconditional ERGMs.

We determine the asymptotic height of the discrete-time critical beta-splitting tree. Let $L_n^*$ denotes the height of the tree with $n$ leaves, defined as the maximum graph distance from the root to a leaf. Then, \[ \frac{L_n^*}{(\log n)^2} \longrightarrow C_{\mathrm{ht}}:=\min_{\theta>1} \frac{\theta}{2\{\psi(\theta)+\gamma\}} \approx 0.976 \] almost surely and in $L^{p}$ for every fixed $p>1$ as $n \to \infty$. Here $\psi$ is the digamma function and $\gamma$ is Euler's constant. This answers \cite[Open Problem~4]{AldousJansonII} of Aldous and Janson.

Motivated by the continuous-time critical beta-splitting tree, we investigate chronological trees recording the successive block splits and their times in homogeneous fragmentation processes restricted to $n$ labels. The separation time of a subset is the first time its labels cease to lie in a common block. For each fixed integer $q\geq2$, we study extremal separation times of $q$-element subsets via their associated point process. Under mild conditions, as $n\to\infty$, the last such time has a randomly shifted Gumbel limit after deterministic centering. Above a threshold $\theta_*$, the coefficients of both the leading $\log n$ term and the $\log\log n$ correction in the centering become independent of $q$, as does the limiting law up to deterministic translation. The extremal process converges to a randomly shifted Poisson point process, without and with decorations, for $q \le \theta_*$ and $q>\theta_*$ respectively. Finally, in the same centered time window, the number of size-$q$ blocks (clades) converges to a pure-death process for $q\leq\theta_*$, and to a process whose trajectories are non-monotone with positive probability for $q>\theta_*$.

Let $A$ be an $n\times n$ symmetric matrix with eigenvalues $\lambda_1\geq\cdots\geq\lambda_n$. Let $\tilde A:=A+E$, where $E$ is a symmetric noise matrix, and denote the eigenvalues of $\tilde A$ by $\tilde\lambda_1\geq\cdots\geq\tilde\lambda_n$. Bounding the perturbation $|\tilde \lambda_i -\lambda_i| $ is a central problem in linear algebra and numerical analysis. In this paper, we prove new perturbation results by exploring the actual interaction between $E$ and the eigenvectors of $A$. In the setting where $E$ does not act adversarially with respect to these vectors (for instance, if $E$ is random), we obtain a considerable improvement over Weyl's inequality. One can routinely extend these results to the Hermitian and rectangular settings. We obtain the new bounds as corollaries of a regional stability result, which provides a sufficient condition for a region on the real line to be stable (containing the same number of eigenvalues) after the perturbation. This result is of independent interest. We prove our regional stability result using contour integral analysis. Our main new technical ingredient here is the double-jump argument, which is robust and could be useful in many other situations involving Neumann series.

The paper focuses on the following problem: given a generalized Bratteli diagram $B$, determine conditions under which the tail equivalence relation $\mathcal R$ does not admit a tail invariant probability measure on the path space $X_B$. We give several sufficient conditions, formulated in terms of the incidence matrices, that guarantee the nonexistence of such a measure. We also discuss a new spectral method for finding tail invariant measures for a class of stationary generalized Bratteli diagrams. This method allowed us to answer affirmatively an open problem formulated in arXiv:2212.13803.

We give a quantitative disorder comparison for the zero-field Sherrington–Kirkpatrick model at inverse temperature one. For any fixed number of replicas, replacing Gaussian couplings by independent couplings from a bounded symmetric variance-one law changes a bounded replica expectation by at most a constant times the Gaussian overlap second moment plus $N^{-1}$. A multiplicative comparison for nonnegative observables controls the tails. The proof uses Talagrand's positive-observable estimates and Yu-Ting Chen's fourth-order replica cancellation. Assuming the Gaussian critical overlap limit stated by Du and Huang, we obtain the same limit for the random quenched overlap measure in the iterated Wasserstein-2 topology. No separate moment assumption is needed for this transfer. Their additional exponential moment bound gives convergence of the rescaled spin-glass susceptibility in every finite Wasserstein distance. The coupling class includes fair signs, uniform disorder, and signs thinned at any fixed positive density.

Graph matrices encode dependencies in random matrices built from shared random variables and arise in spectral algorithms, sum-of-squares (SoS), and high-dimensional statistics. We determine how finite graph structure controls their sharp spectral growth. For every fixed simple graph shape in the dense Rademacher model, including overlapping or empty matrix boundaries, we prove $\mathbb E\|M_\alpha\|=\Theta_\alpha(n^{(v+h-s)/2}(\log n)^{a_*/2})$, where $v$ counts vertices, $h$ isolated summation vertices, $s$ the minimum boundary-separator size, and $a_*$ maximizes an active-component count over minimum separators. Thus two finite cut optimizations determine both the polynomial and logarithmic exponents. The formula closes the polylogarithmic gap in separator bounds, and an infinite family with identical coarse parameters but different norms shows that the logarithmic exponent records genuinely new structure. The proof controls all defect layers in growing trace moments by converting label loss into separator excess; conditional flattening and synchronized fluctuations yield matching lower bounds. We extend the analysis to specified independent-factor chaoses, local weights, unequal dimensions, bounded asymmetric noise, Gaussian inputs, and fixed-degree Hermite inputs. Applications include degree-four clique SoS feasibility for $9\le k\le c\sqrt n$ without an asymptotic logarithmic loss, and Gaussian random tensor networks: deviation thresholds, sharp expected scales, entropy estimates, and, for connected loopless equal-dimensional networks, convergence of the rescaled largest output eigenvalue to the exact right edge of the limiting law. These results connect finite structure to sharp growth scales, and additional algebraic and spectral structure to full feasibility and exact limiting constants.

We develop an exact posterior-sampling framework for Bayesian inverse problems when selected bounded forward observables can be accessed through Bernoulli events. A Bernstein–Poisson construction converts these forward coins into scaled Gaussian likelihood coins, and thinning an inflated prior Poisson point process yields posterior atoms that are iid conditional on their number. For independent Gaussian observations, we derive an exact mean-work identity for the implemented early-stopped factory and sharp small-noise complexity laws governed by local prior-predictive mass near the exact-fit set; a factorial-moment construction extends the likelihood factory to correlated Gaussian errors. The posterior algorithm is model-agnostic once Bernoulli access is available. As one continuum realization, we use Feynman–Kac sampling, Poisson killing, and lazy random-series evaluation for a bounded elliptic resolvent problem with a function-valued coefficient. Numerical experiments validate the forward and likelihood coins, support the predicted work regimes, and demonstrate posterior sampling without deterministic spatial discretization or fixed parameter truncation in the target. A matched-accuracy benchmark against finite-difference prior rejection illustrates how deterministic discretization bias changes the posterior accuracy–cost balance.

We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to $\sigma^2r\log r$ and establish a central limit theorem under the corresponding normalization, where $\sigma^2$ is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.

We resolve a conjecture on the law of large numbers for coherent measures on Young diagrams associated with Macdonald-nonnegative specializations of the algebra of symmetric functions for all $0\le q,t<1$. We prove that row and column lengths divided by the diagram size converge to the corresponding specialization parameters. The proof combines a monomial estimate for skew Macdonald polynomials with exponential tilting and binomial decompositions obtained from the branching rule for the polynomials.

Normalized implied-volatility coordinates reveal no-arbitrage structure that is less transparent in strike space. This paper centers on a single quantity that gives those coordinates a direct probabilistic interpretation.

We prove a lower comparison for disorder-averaged ferromagnetic XY correlations: independent nonnegative random couplings $K_e$ can be replaced by $-\frac12\log\mathbb{E}e^{-2K_e}$. Ginibre's inequality makes each normalized correlation convex in $1-e^{-2K_e}$, so conditional Jensen and thinning compare Bernoulli bond densities; two star-wise Jensen passes handle site dilution on bipartite graphs. Together with the phase-existence theorem of Dario and Garban, these comparisons give local Lipschitz regularity of both the averaged algebraic-phase and susceptibility-divergence thresholds throughout the supercritical density intervals on $\mathbb{Z}^2$. The clean finite-domain criterion of van Engelenburg and Lis also yields explicit finite-size susceptibility lower bounds in a weak-dilution region. For bounded centered bond noise, the possible suppression of the algebraic-phase temperature is at most quadratic in the noise amplitude. Finally, at finite positive thresholds we prove total-variation continuity for bounded iid coupling laws whose essential supremum equals the common support cap, with local Lipschitz control when the reference law has an atom at that cap.

We address two questions posed by Qu and Valkó in their study of the pair correlation function of the $\mathrm{Sine}_\beta$ process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on $(0,\infty)\times\mathbb{R}$, removing the restriction $\beta>2$ in their joint-continuity result. The argument uses a general observation: separate weak continuity of a family of probability laws, together with stochastic monotonicity in one parameter, implies joint weak continuity. Applied to the terminal value of the Qu–Valkó diffusion, their Palm-density formula then gives the result without differentiating the Fourier expansion. Second, for $\beta=2n$ we give a direct proof that the Qu–Valkó matrix-recursion power series agrees with the Selberg-integral representation recorded by Forrester. A Krawtchouk transform converts the Qu–Valkó system into a $(2n+1)$-dimensional differential system with parameter $-(n+1)$, while a centered Aomoto–Selberg trace system has parameter $n+1$. An explicit triangular differential intertwiner reflects the parameter $\eta\mapsto-\eta$. Matching the unique Frobenius branch of exponent $2$ gives exactly the Selberg normalization. As a consequence, every coefficient of the Qu–Valkó recursion is identified with an even centered moment of the corresponding Jacobi–Selberg trace statistic.

The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.

This paper studies the non-stationary marked Hawkes processes in which the base intensity function is time-dependent and the kernel function is governed by an external random factor called mark. The marks are assumed to be determined by random events occurrence over time resulting in a non-identical distribution. These facts introduce two kinds of difficulties: one from the absence of the i.i.d. (independent and identical distributed) and the other is from the non-stationarity. In this framework, an asymptotic regime, often referred to as the high intensity regime, is considered, under which the intensity increases with time. This can be obtained by multiplying the time parameter by n alpha, for some alpha > 0. Under the high intensity regime, the functional law of large numbers and the functional central limit theorem are established. The rescaled (centered and normalized) marked Hawkes process is proved to converge in distribution to a Gaussian process, which is the accumulation of Gaussian noise and a diffusion process. In addition, a shot-noise process is studied, and the functional Limit Theorems under appropriate hypotheses are established.

In this paper we study local uniform, with respect to time, asymptotic expressions for the asymptotic behavior of the entrance probability of discounted aggregate claims to some rare sets, in a multivariate risk model with arbitrarily dependent insurance and financial risks. Our model is based on a multivariate version of a model, introduced by Guo (2022), and we consider that the logarithmic return process of the insurers investment portfolio is described by a jump-diffusion process. The dependence between the insurance and financial risks is implied by the dependence of the claim-vectors with the jumps of returns, and is arbitrary under some distributional conditions on the claim-vectors and the discounted claim-vectors. In opposite to previous papers on this topic, except the multidimensional extension, we consider that the model is driven by a common counting process, that is not necessarily renewal, the claim vectors are interdependent, and their distribution is not restricted to the class of (multivariate) regularly varying distributions. Under the condition that the claim vectors, and the discounted claim-vectors follows distributions from the class of multivariate consistently varying and positively decreasing distributions, and under some moment conditions on the jumps and the counting process, our main result shows the presence of multivariate linear single big jump principle of the discounted aggregate claims in this not necessarily Levy-Renewal environment. After restriction of the distributions to multivariate regular variation, we obtain more explicit expressions, under a slightly weaker moment condition on the jumps. We provide a corollary, in which the conditions of the main result are satisfied under a weak dependence structure and we find a more explicit asymptotic expression, using the technique of solution of the dependence.

We construct a product spectral decomposition from power-normalised parabolic-cylinder functions. At noninteger real orders, these functions are not square integrable with respect to the corresponding full-line Gaussian measure. Bringing the admissible half-line branches into the same Gaussian space and matching them at the origin gives a corrected orthonormal eigenbasis. A unitary transformation to a weighted probability space makes the eigenfunction corresponding to the lowest eigenvalue constant and permits consistent countable products. The resulting law is non-Gaussian under deformation, and the decomposition recovers classical Wiener chaos when the deformation is removed. We identify the associated closed gradient, adjoint divergence and deformed number operator. The Rodrigues formula covers all orders occurring in the spectral branches, including negative orders. Under deformation, however, changing both branch orders by the same nonzero amount breaks the required relation between them. Fractional evolution defined through spectral calculus preserves the basis. After unitary identification, the one-coordinate fractional evolution operators converge in operator norm to Gaussian and three-dimensional radial Ornstein–Uhlenbeck fractional evolutions at the respective parameter endpoints. Convergence is uniform on compact time intervals away from zero. An application example with inverse stable clock is provided in the accompanying github repository.

Consider the heat equation: \begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} It is well known that superlinear growth of the coefficient $b$ may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient $b:\R\to\R$, there always exists a locally Lipschitz noise coefficient $\sigma:\R\to\R$ such that the stochastic heat equation \begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt+\sigma(u(t,x))\,dB_t, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} has a unique global solution for every $u_0\in C_c(D)$. Here $(B_t)_{t\ge0}$ is a real-valued Brownian motion. In particular, if $b=0$, the equation has a global solution for every locally Lipschitz $\sigma$, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.

We derive a first-order representation of Black-Scholes implied variance in a continuous local martingale model. Total implied variance is the conditional expectation of the quadratic variation of the log price given its terminal value, up to a smaller-order term, for bounded standardized log-strikes. The framework incorporates small volatility-of-volatility, fast mean-reverting, and short-maturity asymptotics.

Let $Y_1,\ldots,Y_n$ be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process $X_x=\sum_{i=1}^n x_iY_i$ without any $\Delta_2$ or regular-growth assumption on the coordinate tails. The characterization is governed by scale-dependent intrinsic costs determined directly by the convex tail potentials. It has an information-theoretic rate-distortion formulation as well as two equivalent majorizing-measure formulations on the original index set, one multiscale and one based on a single probability measure. More strongly, the comparison holds for every prescribed law of the index. In particular, the Gaussian majorizing-measure theorem of Talagrand and the Bernoulli theorem of Bednorz-Latała are recovered from the same formula, through quadratic and truncated quadratic costs, respectively. The new geometry is necessary: once the $\Delta_2$ growth condition is removed, the usual chaining functional based on increment moments can exceed the expected supremum by an unbounded factor. The proof is based on a uniform convex-order truncation principle for symmetric exponentials and a dyadic decomposition of the tail slopes that transfers simultaneously the random process and its intrinsic cost, with universal constants.

The Swing filter extends a feasible line through its previous recording and restarts from a fitted value at the last accepted sample. We show that this inherited endpoint causes a singular loss of compression on rough input. For uniformly sampled Brownian motion at fixed tolerance, every fixed number of recording times converges to the first continuous feasibility boundary, with alternating boundary errors. The longest post-first segment tends to zero in probability. The segment-count ratio against a continuous approximation through true sampled knots diverges almost surely whenever that boundary precedes the observation horizon. For unit Gaussian increments, constrained least squares and no maximum lag, the first mean segment length has quadratic order in the tolerance $E$, the second has linear order, and every fixed mean from the third onward has order $E\log E$. We explain the logarithm through the conditional survival tail and prove a geometric bound on the subsequent approach to the stationary coefficient in the large-tolerance limit. The stationary mean satisfies $C_E\sim\gamma_{\mathrm{Swing}}E\log E$. An explicit one-dimensional kernel specifies the single coefficient $\gamma_{\mathrm{Swing}}\approx1.8120703$ and admits a $3/4$ Wasserstein contraction and convergent evaluation brackets. The decimal is not a certified error interval. A half-budget slope corridor, true endpoint chords and one-sample bridges repair the degeneration while preserving the original error bound, continuous output and constant working memory. Including bridge costs, the long-stream mean span per output segment is asymptotic to $21\zeta(3)E^2/(8\pi^2)$. Its explicit coefficient follows from the known Brownian anchored lifetime. Thus a change to the recording rule restores the quadratic scale.

We introduce a notion of viscosity solution for Hamilton–Jacobi–Bellman (HJB) equations with distributional drift, based on paracontrolled test functions and related through a Zvonkin transformation to classical viscosity theory. The equations considered are of the form \[ \left(\partial_t+\frac12\Delta+b\cdot\nabla\right)h(t,x) =-H(t,x,h(t,x),\nabla h(t,x)), \] where $b$ is singular in the sense of \cite{paradistrib} and has regularity $\mathcal C^{-\alpha}$ for $\alpha\in(1/2,2/3)$. Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in $\nabla h$. We also obtain probabilistic representations through singular control problems for convex $H$ and weak singular forward–backward SDEs for possibly nonconvex Hamiltonians with at most quadratic growth.

We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

We study the spherical statistics of all the points in a thin shell of a high-dimensional random lattice. The exact relations between lattice points make it unclear when predictions based only on spherical geometry should hold. We answer this question for several statistics, including the number of shell points, the balance of their directions, and the occurrence, repetition, and distribution of differences between them. We identify sharp thresholds as the shell radius grows and show that these statistics undergo several distinct phase transitions. A shell can already agree with one geometric prediction while still differing strongly from another. Our results hold for a single sampled lattice, with probability tending to one as the dimension grows. They include estimates that hold across a complete shell, bounds close to the transition thresholds, and extensions to randomly shifted lattices. A Lean formalization verifies the main results, assuming the classical formulas and probability model stated in the code.

We prove the sharp $\ell^p$ bound $p^\star-1$ for certain second-order discrete Riesz transforms on products of integers, in all dimensions. The family includes the imaginary part of the discrete Beurling-Ahlfors operator in dimension two as a special case. While these operators admit a martingale representation via compound Poisson jump processes and heat extensions, the martingale attached to a function and the martingale attached to its transform do not enjoy strong differential subordination, so the operator-norm estimate is not accessible through prior work.

We study the mixing behavior of three q-deformed card shuffles on $\mathcal{H}_q(S_n)$: the short systematic scan introduced in \cite{DiaconisRam2000}, the $q$–deformed random–to–random shuffle introduced in \cite{AxelrodFreedBraunerChiangComminsLang2024}, and the $q$–deformed $k$–star transposition shuffle, whose $q=1$ counterpart was studied in \cite{ArfaeeNestoridi}. The first two chains were diagonalized in \cite{DiaconisRam2000} and \cite{AxelrodFreedBraunerChiangComminsLang2024}, respectively. Here, we diagonalize the $q$–deformed $k$–star transposition shuffle and use the spectra of the three chains to study their mixing behavior. For fixed $q>1$, we prove that all three exhibit total variation and $\ell^2$ cutoff. For the short systematic scan, these improve both the upper and lower bounds of Diaconis and Ram \cite{DiaconisRam2000}.

Statistical validation of synthetic multivariate data requires assessing whether a generator preserves the joint dependence structure of the target population without merely reproducing observed records. We develop a model-agnostic framework based on full conditional distributions. For each coordinate, we normalize the conditional probability assigned to the observed value by the largest conditional probability available in the same record context; averaging this quantity yields a one-sided MAP-alignment statistic that can be estimated using a conditional model fitted on held-out real data. The mathematical contribution is twofold: under strict positivity and compatibility, the complete normalized conditional profile identifies the joint distribution, and its integrated L1 difference defines a metric on finite-state generative processes; we also establish consistency and finite-sample concentration for the corresponding empirical estimators. Because high conditional alignment alone can arise from copying or concentration on conditional modes, we pair it with nearest-real similarity as a separate record-level novelty diagnostic. We evaluate the framework on NSHAP health and aging data, influenza B genomic surveillance, and 34 General Social Survey waves. In GSS, the Large Science Model matched the original-data control in mean conditional alignment while retaining substantial novelty, indicating preservation of conditional structure without row reuse. In influenza B, a Chow-Liu generator matched the control alignment but had almost no novelty, revealing near-reproduction of observed records. The framework therefore distinguishes three statistically different failure modes: loss of dependence, record reuse, and mode concentration, and provides a principled basis for validating synthetic health, surveillance, and population data.

The basic question in perturbation analysis of Markov chains is how small changes in their transition kernels affect their stationary distributions. Classical perturbation bounds typically require the kernel error to be much smaller than $1/\tau$, where $\tau$ is a mixing or relaxation time. Although this scaling is sharp for Markov chains in general, we investigate a general "square-rooting" phenomenon in which one-step errors of order roughly $1/\sqrt{\tau}$ can be sufficient for local updates. We proved a form of this phenomenon in Lin, Liu and Smith (2025) under strong assumptions. Here we substantially weaken these assumptions, and prove this phenomenon occurs using three distinct approaches. First, block factorization applies under structural assumptions on the stationary measures of both chains. Second, approximate block-update arguments extend the result to statistically relevant Markov chain Monte Carlo (MCMC) settings, where structural guarantees are available for the exact posterior and its associated sampler, but not for the perturbed posterior. Third, we use direct calculations for a class of models with hard constraints where neither general result is directly available. We illustrate these results in three MCMC settings and show how they directly inform the tuning of approximate MCMC algorithms.

We study the mixing time of a systematic-scan analogue of Kac's walk that was proposed as a fast surrogate for Haar-distributed orthogonal matrices in randomized high-dimensional algorithms and was conjectured to approach Haar measure after only logarithmically many sweeps. We show that this conjectured speed-up does not occur for convergence of the full matrix law to Haar measure in Frobenius Wasserstein distance. At fixed normalized accuracy, the mixing time lies between order $n/\log n$ and order $n$ sweeps; at fixed absolute Frobenius accuracy, the corresponding bounds are between order $n$ and order $n\log n$. More strongly, below the scale $n/\log n$, the normalized Wasserstein distance remains asymptotically at its extremal value. We also show that the output law is singular with respect to Haar measure for fewer than $n/2$ sweeps. Thus the sampler may provide effective application-specific randomization without exhibiting the much faster full-Haar mixing.

In this paper, we study a stochastic Susceptible-Infected-Removed (SIR) model where the infection and the recovery rates depend on individual covariates for susceptibility and infectiousness of the infector and the infectee. Such models allow explicit nonlinearity in the incidence term. They are also important from a practical perspective, as they allow for the incorporation of individual heterogeneity into the epidemic process. Statistical estimates for crucial epidemiological parameters, such as the basic reproduction number, herd immunity threshold, could be vastly different, and even biased, when the population heterogeneity is ignored in the mathematical model. We describe our epidemic model as an Interacting Particle System (IPS) of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures. Our main mathematical contributions are a Functional Law of Large Numbers (FLLN), which approximates the empirical random measure of the IPS by means of a deterministic measure-valued function, and the propagation of chaos phenomenon, which establishes asymptotic independence of the particles as the population size goes to infinity with an explicit construction of McKean–Vlasov type Kac's “nonlinear process”. We also briefly mention how the propagation of chaos phenomenon leads to a product-form likelihood function, which forms the basis of the so-called Dynamic Survival Analysis (DSA) method for parameter inference based on sparse data.