Papers

We investigate Strong Data-Processing Inequality (SDPI) constants for Rényi Divergences on finite spaces. We study their dependence on the Rényi order $\alpha$, proving that they are non-decreasing and that their scaling by $(\alpha-1)$ is convex for $\alpha\geq1$. We also identify several support restrictions on the probability measures involved in determining these constants. In particular, in the distribution-independent setting, measures supported on a common set of at most two points suffice to evaluate the Rényi-SDPI constant. For $\alpha\in[0,1]$, we further prove equality with the $\chi^2$-SDPI constant, while at order infinity we obtain a closed-form expression. In order to link contraction over product spaces to contraction along individual coordinates, we provide tensorisation bounds for arbitrary product channels analogous to those known for $\varphi$-Divergences. At finite orders, the Rényi-SDPI constants are bounded above and below through comparisons with the $\chi^2$-Divergence and Hellinger Divergences, with sharpness established in multiple cases. At order infinity, we instead relate these constants to the contraction of Total Variation Distance. Finally, our findings are applied to local differential privacy (LDP) and the analysis of Markov chains. This yields sharp contraction guarantees for pure-LDP mechanisms, and connects Rényi-LDP to Rényi-SDPI constants. For Markov chains, we derive finite-time convergence bounds and exhibit a family of chains for which Rényi-SDPIs improve on classical $\chi^2$-based bounds by arbitrarily large factors.

This thesis studies how disorder and fluctuations shape large-scale growth phenomena in three related settings: fluctuating interfaces, random spring chains, and random multiplicative growth. The first part concerns interface growth and a phenomenon denoted anomalous scaling, where local and global exponents differ.To clarify its origin, we study two models in which this mismatch has different causes. The first is a heterogeneous elastic line, where we show that the apparent anomaly is a purely statistical effect. The second is the stochastic porous medium equation, a strongly nonlinear model in which the anomaly is genuine. The second part concerns Anderson localization, which appears here through the spectral properties of the heterogeneous elastic lines. We study chains with both random masses and random spring constants, with particular emphasis on the strong-disorder regime, where standard weak-disorder expansions break down. We develop a new combinatorial method to probe this regime. The last part concerns random multiplicative growth with redistribution. In such models, changes in wealth, population, or mass, for instance, are proportional to the amount already present, so that small differences are amplified over time. This naturally generates broad distributions and provides a simple mechanism for the emergence of inequality and concentration, with applications to population dynamics, wealth distribution, city growth, and ecology. We in particular study models in which each site has its own quenched growth rate, representing a persistent advantage or disadvantage, and is also subject to transient fluctuations. Redistribution competes with these mechanisms by homogenizing the system. This competition leads to rich phase diagrams.

Motivated by the structure of various standard statistical arrays, Bailey, Praeger, Rowley and Speed (1983) described the automorphism groups of poset block structures as generalized wreath products. We observe that these are exactly the groups used to define DAG exchangeability in Jung, Lee, Staton and Yang (2021), and we restate the representation theorem of the latter work in this language. As a concrete example, we write out the resulting representation for block matrices whose block rows and block columns are separately exchangeable, and whose rows and columns may also be permuted independently inside each block.

We study a one-dimensional nonlinear stochastic wave equation driven by Gaussian noise that is white in time and rough in space. We prove a bi-parameter local linearization for mixed increments along the two characteristic directions. The proof combines characteristic cancellation with a localized fractional-energy estimate that controls boundary interactions caused by the rough spatial noise. As an application, we establish quadratic-variation limits on arbitrary anisotropic rectangular meshes and construct a consistent estimator of a multiplicative diffusion parameter. Numerical experiments illustrate the finite-sample performance of the estimator.

We study the Ising spin glass with independent, identically distributed couplings on the diamond hierarchical lattice with $n$ branches, of effective dimension $1+\log_2 n$; $n=4$ is the Migdal-Kadanoff lattice of $\mathbb{Z}^3$. For $n\ge4$ and every coupling law with a bounded density we prove that at sufficiently low temperature $T>0$, uniformly in the level of the lattice, the poles are ordered, every spin has Edwards-Anderson parameter $1-O(T)$ when the poles are fixed, and the overlap of two free-boundary replicas has second moment $1-O(T)$. For standard Gaussian couplings, $n=3,4,5,7,8$ and every $T\le T^*(n)=4/15,\,6/7,\,6/5,\,7/4,\,2$ ($0.89$ to $0.98$ times numerical estimates of $T_c$), a computer-assisted proof gives pole order and, as the level tends to infinity, site-overlap order and the blue-cluster statements below; likewise for every larger $n$ at these temperatures. Previously, spin-glass order on these lattices with frustrated independent couplings had been proved only for more than $10^{18}$ branches (Collet and Eckmann, 1984). The key step is growth of the effective coupling under the exact renormalization map: for $n\ge4$, by a one-step contraction, via Rogozin's inequality, of a density bound on an event of high probability; near $T_c$, by interval-arithmetic bounds in the peakedness order. A transfer theorem following the ancestor-chain argument of Collet and Eckmann turns this growth into bulk statements. In the Chayes-Machta-Redner representation the same input gives a blue cluster that contains both poles with probability tending to one and whose expected density is bounded away from zero, while the largest other blue cluster has density tending to zero in probability and in $L^1$; hence blue-cluster density imbalance. To our knowledge this is the first proof of such an imbalance for a spin glass on a graph other than the complete graph.

Many economic applications establish stability of Markov dynamics using the monotone mixing condition (MMC) of Hopenhayn and Prescott (1992). Working in the same setting as that paper, we introduce a weak monotone mixing condition (WMMC), phrased in terms of the reversal of rank between populations, and show that it is strictly weaker than the MMC and both necessary and sufficient for global stability under the Kolmogorov metric. We apply these results to a small open economy version of the Aiyagari model, showing how the WMMC can be used to establish global stability in a setting where the MMC fails. In addition, we show that, when the state space is one-dimensional, the MMC and WMMC coincide, implying that the original MMC is necessary as well as sufficient in this setting.

We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian $\Delta$ on the unbounded Vicsek set. The space carries its Hausdorff measure $m$. The gradient $\partial$ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure $\nu$ that is singular with respect to $m$. The relevant fractional order for the Riesz inequalities is not $1/2$ but $\gamma_{p}=\frac{d_f+p-1}{pd_w}$, where $d_f=\log_{3}5$ and $d_w=d_f+1$. For $p\in[1,2]$, we prove the weak-type reverse Riesz inequality $\|(-\Delta)^{\gamma_p}f\|_{L^{p,\infty}(m)}\lesssim\|\partial f\|_{L^p(\nu)}$. The corresponding strong estimate fails for $1\le p<2$, whereas the weak estimate fails for $2<p<\infty$, even after heat regularization. For the Riesz transform $\mathcal{R}_p=\partial(-\Delta)^{-\gamma_p}$, we show that $\mathcal{R}_1$ is not of weak type $(1,1)$ but that $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^{p,\infty}(\nu)$ for $p\in(1,2)$, and from $L^{p,1}(m)$ to $L^p(\nu)$ for $p\in(2,\infty)$. However, if $p \ge 1$, $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^p(\nu)$ only for $p=2$.

A submetric space is a topological space equipped with a continuous metric that generates a metric topology weaker than the original one (e.g., a separable Hilbert space with the weak topology). In this paper, we systematically investigate the compactness-related properties of submetric spaces, which make this class of topological spaces a particularly suitable tool for probabilistic applications. As an illustration, we discuss in detail the compactness of the closure of a càdlàg trajectory taking values in a submetric space.

We study the long-time dynamics of a class of singular stochastic partial differential equations with multiplicative Gaussian noise. Using the theory of regularity structures, we construct the associated random dynamical system and establish an ergodic framework for the underlying random models. Combining pathwise estimates for the singular equation with a multiplicative ergodic theorem in Banach spaces, we analyse the Lyapunov spectrum of the linearized dynamics and establish the existence of local stable, unstable, and center invariant manifolds around stationary points. As a consequence, when the linearized equation is exponentially stable, we obtain a local exponential stability result for the nonlinear equation.

Graph learning from edge-detecting queries concerns the reconstruction of an unknown edge set on a known vertex set. Each query reports whether a specified vertex subset contains at least one edge. We study non-adaptive schemes, in which all queries are fixed before any outcomes are observed, with the goal of achieving exact recovery using few queries and fast decoding. For general graphs on $n$ vertices with at most $k$ edges, non-adaptive recovery requires $\Omega(\min\{k^2\log n,n^2\})$ queries in the worst case, even when a small error probability is allowed. In this paper, we consider Erdős–Rényi ($\mathrm{ER}$) graphs $G\sim \mathrm{ER}(n,q)$, with expected edge count $\bar{k}=q\binom{n}{2}$. Our scheme uses $O(\bar{k}\log n)$ queries and achieves exact recovery in $O(\bar{k}\log n)$ decoding time with probability tending to one throughout the regime $\bar{k}\to\infty$ and $\bar{k}=o(n^2)$. This improves the previous $O(\bar{k}^{1+\delta}\log n)$ decoding guarantee for any fixed $\delta>0$, while maintaining the same query order. The guarantee also extends beyond the previously studied regime $\bar{k}=\Theta(n^{2\theta})$ with fixed $\theta\in(0,1)$. Our approach builds on the binary splitting method used in prior work, which organizes vertices into a hierarchy of successively smaller groups. We introduce three main changes: (i) we use random affine hash functions over a finite field to process each candidate pair in constant time; (ii) we apply the splitting procedure directly to the full graph, avoiding the need to combine solutions to multiple smaller graph-learning subproblems; and (iii) we bound the total decoding workload directly rather than deriving separate high-probability bounds on candidate counts at each level.

In an earlier paper, M. Talagrand made a number of daring conjectures. An AI model provided a very beautiful proof of one of them which we explain here.

We introduce the $m$-ary recursive tree, a tree structure that starts with a single node. At each step of the growth process, a random node, called the recruiter, is selected, and $m$ new nodes are attached to this recruiter. The main objective is an asymptotic analysis of distances in this structure. We first derive the distribution of the depth of nodes in such a tree as a convolution of independent (though not identically distributed) Bernoulli random variables. This representation yields normal and Poisson approximations with quantified errors for the depth. We also investigate the height of these trees, that is, the longest root-to-leaf distance, via a coupling to a weighted random recursive tree with suitably chosen weights. The coupling transfers the results on a strong law and tightness from the weighted recursive tree to the $m$-ary recursive tree. Furthermore, we establish a bivariate limit law for the Wiener index, that is, the sum of all pairwise distances, and the internal path length, that is, the sum of the depths of all nodes, using the contraction method. As a novel technical feature of our approach, we employ weighted norms instead of Euclidean norms within the contraction method.

Metric graphs are generalisations of linear networks and provide a natural framework for the definition of continuously-indexed Gaussian processes. We define a new class of distances on these topologies, termed polyharmonic distances, which unify and extend the spectral construction underlying the effective resistance distance and the biharmonic one. We give both a spectral and a variational characterisation. Furthermore, we show an explicit class of stochastic processes whose variograms coincide with the squared polyharmonic distances. Finally, we show how these metrics can be composed with suitable completely monotonic functions to define isotropic processes having any prescribed finite-order mean-square differentiability along the edges and satisfying the Kirchhoff conditions up to order one at the vertices.

Locally stationary Markov chains have attracted a lot of attention in statistics in the past three decades. In this paper we prove a variety of limit theorems for partial sums generated by such chains. We first provide explicit formulas for the asymptotic mean and variance (and also higher moments), and obtain optimal convergence rates towards them. We then prove a Berry–Esseen theorem, a local central limit theorem, Edgeworth expansions, and large and moderate deviations principles. Our approach involves a parametric Perron–Frobenius theorem, which is proved using the theory of complex (Hilbert) projective metrics developed in \cite{Rugh,Dubois}, together with local approximation arguments and ideas in \cite{dolgopyat2023berry}.

We study robust reconstruction for the \(q\)-state Potts broadcast process on a Galton-Watson tree in the growing-alphabet regime \(q\to\infty\), with boundary depth growing sufficiently quickly in relation to \(q\). We show that, in the regime \(d\lambda >1\), where \(d\) is the expected number of offspring and \(\lambda\) is the nontrivial eigenvalue of the Potts transition matrix, the root posterior is asymptotically unchanged by a broad class of noise channels applied independently to the boundary labels. This remains true even when the probability of retaining the true label at an individual boundary vertex tends to zero as \(q\to\infty\). More precisely, under suitable moment and noise assumptions, the noiseless and noisy root posteriors converge to one another in expected total variation, and hence have the same asymptotic Bayes-optimal reconstruction accuracy.

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|\delta V\|_{L^p(\gamma_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(\gamma_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2\mathcal N^{-1}$, where $\mathcal N := -\Delta+x\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\geq2$, and $m_p := (\mathbb E|g|^p)^{1/p}$ for a standard Gaussian $g$, then \[ \|DP\|_{L^p(\gamma_d;\mathbb R^d)} \leq\frac{2\sqrt{2e\, \mathrm{deg}(P)}}{m_p}\,\|P\|_{L^p(\gamma_d)}\, . \] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\sqrt{\mathrm{deg}(P)/p}$ throughout the range $p\geq2$.

We compute $\E[(S_T-K)^+]$ by Monte Carlo for a scalar stochastic-volatility model with a fast mean-reverting factor of time scale $\eps$, for $\eps$ ranging from $1$ down to $10^{-3}$. A conditional (mixing) estimator gives finite variance, whereas the direct estimator has infinite variance for this model. The volatility factor is simulated with its exact Ornstein–Uhlenbeck transition. As $\eps\to0$ the price converges, at rate $O(\eps)$, to the Black–Scholes price with the averaged volatility $\bar\sigma$, and the implied-volatility smile flattens to $\bar\sigma$. Finally, we test a martingale control variate built on the Black–Scholes delta with volatility $\bar\sigma$. If the martingale is driven by the true volatility $\sigma(Y_t)$, the variance is reduced by a factor that grows like $1/\eps$, about $160$ at $\eps=10^{-3}$. If it is driven by the constant $\bar\sigma$, the variance is essentially not reduced. Last, we calibrate the model with $\rho\neq0$ to S&P~500 implied volatilities by full simulation of $S$. At the same number of paths the control variate reduces the variance by a factor $1.7$–$16$ (median $5.6$) and gives more accurate calibrated parameters; at equal CPU time it pays off only if the delta is rebalanced on a coarser grid than the time step. For a basket of four indices (state dimension $8$) the gain is larger (median $8$) and the relative cost smaller, so that the control variate is about $4$–$5$ times more efficient than plain Monte Carlo at equal CPU time.

Domain Adaptation (DA) often lacks worst-case guarantees, while Distributionally Robust Optimization (DRO) based only on source data centers its ambiguity set at the source law and ignores available target structure. To bridge this gap, we introduce a doubly-anchored DRO framework whose ambiguity set is the intersection of $\phi$-divergence balls centered at the source law and a source-completed target reference law, the latter pairing the target covariate law with the source conditional law. We derive dual-induced adversarial bridge geometries for symmetric and asymmetric divergence pairings, notably introducing a Kullback–Leibler/squared-Hellinger (KL/HD) bridge. This asymmetric formulation yields a Lambert-$W$ geometry in which source-side exponential risk tilting and target-side Hellinger stabilization enter through distinct terms, attenuating, but not bounding, the effect of large likelihood ratios. Furthermore, without imposing covariate shift, we establish finite-sample generalization bounds for the minimizer of a structural, loss-agnostic density-bridge risk under bounded-overlap conditions; these bounds do not apply directly to the loss-aware DRO min–max estimator. We translate our framework into a bridge-weighted Nadaraya–Watson estimator, proving uniform consistency for the source regression function and pointwise asymptotic normality, with target recovery when the source and target regression functions coincide, as under covariate shift. Finally, an empirical evaluation on a domain-shifted Fashion-MNIST dataset illustrates the finite-sample stability of the asymmetric KL/HD bridge under severe synthetic target-covariate corruption.

We study plug-in estimators of the entropy rate and the transfer entropy rate for stationary finite-alphabet chains with unbounded memory. Under a coalescent coupling-from-the-past assumption, we establish non-asymptotic concentration inequalities around the expectations of the estimators. Under additional uniform non-nullness and exponential beta-mixing assumptions, we control the corresponding biases and obtain concentration around the entropy and transfer entropy rates, together with almost-sure consistency. The results apply to chains governed by g-functions that need not be globally continuous.

We identify the limit free energy of all mean-field multi-species spin glasses defined on products of spheres. In particular, we do not make any convexity assumption on the covariance function of the Hamiltonian, and we allow for the presence of external fields. We also obtain a one-species reduction of the formula in the special case of balanced models.

We study how the affine geometry of finite ReLU networks is created at random initialization and reorganized by supervised training. We call a sign-changing zero of a hidden preactivation an activation switch and a point where the scalar network output is nondifferentiable a scalar kink. For one-dimensional input, conditioning on the preceding layers makes each preactivation Gaussian and affine on the cells of a random finite partition. This yields an exact finite-width conditional Kac-Rice formula for the expected number of activation switches along an input interval. For fixed depth and proportionally growing widths, the resulting switch intensities converge to explicit deterministic limits. A visibility estimate shows that the expected number of switches that do not produce scalar kinks is negligible, yielding an explicit leading formula for the expected number of scalar kinks and hence affine regions. In higher input dimensions d >= 2, the analogous conditional surface formula yields the leading expected (d-1)-dimensional Hausdorff measure of the scalar kink set. On the Breast Cancer Wisconsin data, the initialization formula accurately predicts switch counts along held-out segments. After training, switch counts decrease along within-class segments and increase along between-class segments. Thus, training redistributes rather than merely contracts affine complexity.

For a class of non-stationary linear processes $(X_{n})_{n}$ with weakly dependent heavy-tailed innovations we prove a convergence result for the space-time point processes $N_{n}=\sum_{i=1}^{n}\delta_{(i/n, X_{i}/a_{n})}$. Using this result and the continuous mapping theorem we then derive functional convergence of the corresponding partial maxima process in the space of càdlàg functions on $[0,1]$.

This paper addresses fundamental questions about the existence and structure of isotropic and anisotropic spherical random fields and the properties of composite transformations of isotropic fields. It introduces a hierarchical structure comprising several classes of anisotropic random fields, using geometric and spectral approaches, and provides a geometric–spectral classification of second-order anisotropy on $\mathbb{S}^2$. We characterize feasible isotropic and anisotropic scenarios and identify those that, in principle, cannot occur, in contrast to the case of random fields in Euclidean space. Furthermore, we establish connections between geometric invariance properties and the structure of the spectral covariance matrix of spherical random fields. The obtained results lay the foundations for applications in statistics and data analysis, providing guidance on feasible models, their spectral representations, and appropriate sampling schemes for parameter estimation or anisotropy hypothesis testing for spherical data.

In this note, we consider the advection-diffusion equation on the $d$-dimensional torus, where $d$ is arbitrary. First, for the associated transport equation, we provide an explicit example of a space-time smooth, divergence-free, random velocity field that is a universal exponential mixer. Second, we set the molecular diffusivity to a (small) positive value, and we show that the same velocity field is dissipation enhancing with optimal rate. Our construction is based on the Random Dynamical Systems viewpoint, and relies on the uniform ergodicity of the two-point Markov process associated with the two-dimensional Pierrehumbert model.

Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $\gamma_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely the cube $[-1,1]^n$, can be recovered from its values at the vertices. For $P\in\mathcal{P}_m(\mathbb{R}^n)$, set \[ M(P)=\max_{x\in[-1,1]^n}|P(x)|, \qquad V(P)=\max_{\varepsilon\in\{-1,1\}^n}|P(\varepsilon)|. \] If $P_n$ is chosen according to $\gamma_{m,n}$, we prove that the relative loss \[ 1-\frac{V(P_n)}{M(P_n)} \] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<\beta<1/2$, \[ \gamma_{m,n}\left\{ P\in\mathcal{P}_m(\mathbb{R}^n): V(P)\geq (1-n^{-\beta})M(P) \right\} \longrightarrow1 \] as $n\to\infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.

We study the survival of a branching random walk in a supercritical Galton-Watson tree subject to a deterministic, superlinear killing barrier with heavy-tailed displacements. For a fixed $m>0$, we consider a family of such barriers and kill particles whose ancestral paths fall below the barrier. Under some standing assumptions, we establish a sharp phase transition at $m=4$: the process becomes extinct almost surely for $m<4$, while survives with positive probability for $m>4$. Notably, a natural first-moment heuristic suggests the threshold $m=2$; the true critical value $4$ arises from a deterministic constraint along infinite surviving rays. At the critical value $m=4$, we provide examples showing that the same standing assumptions do not determine the survival behavior. The proofs are based on analysis of infinite surviving rays and embedded trees.

In sampling, different flavours of annealing, tempering, or other successive approximation approaches are widely used and studied. In this work we investigate annealed underdamped Langevin sampling for logconcave distributions of the form $\pi(x) \propto e^{-\Psi(x)}$ for a potential $\Psi:\mathbb{R}^d \rightarrow \mathbb{R}$. That is, we assume access to the score of a general approximating family of distributions $(\pi_{\tau})_{\tau\in [0,1]}$ with $\pi_0=\pi$. This family is used in a time-inhomogeneous underdamped Langevin process with moving target $\pi_{\tau(t)}$ and an appropriate annealing schedule $t\mapsto \tau(t)$ such that $\tau(t)\downarrow 0$ as $t\to T$ for some $T>0$. While the degeneracy of the Brownian motion prohibits conventional Girsanov arguments, we are able to circumvent this issue by relying on an appropriately designed auxiliary process. To the best of our knowledge these are the first (quantitative) results for sampling using time-inhomogeneous underdamped Langevin dynamics. Moreover, we obtain an iteration complexity of $\mathcal{O}(\epsilon^{-2})$ in total variation distance in comparison to $\mathcal{O}(\epsilon^{-6})$ for annealed overdamped Langevin sampling. While we currently assume convexity, our proof strategies are amenable to generalization for non-logconcave targets and enable for several future research directions.

This paper is concerned with the maximal regularity theory for non-autonomous stochastic evolution equations with a generalized fractional derivative in UMD spaces. The generalized time-fractional derivative provides a unified framework covering both the classical Riemann-Liouville and Caputo fractional derivatives, which accommodates a wider class of anomalous diffusion processes with intermediate memory effects. Based on the singularities of the initial term and the stochastic convolution kernel, a time-weighted space and a regular-singular decomposition are used to obtain the well-posedness, space-time regularity, and stochastic maximal $L^p$-regularity results. Our results are applied to non-autonomous stochastic diffusion equation and stochastic fractional reaction-diffusion SIR model.

We study equilibrium joining strategies in an M/M/1-type queueing system with strategic customers operating in a two-phase random environment described as a continuous-time Markov process. Strategic customers, upon arrival, choose whether to join or to balk based on available information and anticipated utility, considering the trade-off between reward from service and waiting cost. Four observation scenarios are analysed: fully observable (both queue length and environment phase are disclosed to a customer upon arrival), queue-only observable, environment-only observable, and fully unobservable. In each case, equilibrium joining strategies are analysed. In the environment-only observable and fully unobservable cases, explicit solutions and equilibrium conditions are derived under rapid oscillations and under very slow transitions between environment phases.

The quantum baker map is a standard model of quantum chaos, but rigorous analysis of how it spreads random perturbations remains difficult. Following the repeated “baker plus refreshed diagonal perturbation” architecture of Schack and Caves, we study a tractable strong-noise model based on the Balazs–Voros quantization: at each time step, its unitary propagator is followed by a diagonal unitary whose entries are independent and uniform on the unit circle. The resulting random propagator is a structured random walk on $\U(N)$, $N=2^k$. We quantify how quickly the baker dynamics spreads this basis-local randomness at three levels: ensemble means, two-copy statistics, and the full law of the accumulated propagator. The walk becomes an exact unitary $1$-design at time $k+1$, while its fixed-accuracy $2$-design mixing time is $\Theta(\log N)$. Thus the induced state ensembles reproduce Haar means and two-copy statistics on the Ehrenfest scale. The full law mixes much more slowly. A phase-adapted path coupling gives normalized Wasserstein mixing in $O_\varepsilon(N)$ steps, whereas metric-entropy and Haar small-ball estimates give an $\Omega_\varepsilon(N/\log N)$ lower bound and asymptotically maximal distance at every $t=o(N/\log N)$. The law remains singular with respect to Haar measure for every $t<N$. The moment results and the Wasserstein upper bound are driven by the unistochastic Markov kernel associated with $B_N$, an affine dyadic chain whose nonconstant Fourier modes vanish exactly after $k$ steps. The resulting hierarchy separates Ehrenfest-scale randomization of state ensembles from near-Heisenberg-scale exploration of the full unitary group.