Papers

We develop a quantitative framework for the long-time behavior of McKean–Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition with a one-level trapping mechanism, which requires dissipativity at only one admissible moment level rather than global contraction. This permits distribution dependence with supercritical growth and thus is more compatible with phase-transition models. For uniqueness and convergence, we establish an anchored uniqueness and quantitative ergodicity principle. An explicit \(L^1\)-smallness condition, expressed through a convolution kernel built from derivative estimates of the anchored frozen semigroup, yields uniqueness and transfers exponential or polynomial mixing rates of the frozen dynamics to the nonlinear McKean–Vlasov system. Moreover, the principle is sensitive to the choice of topology: different distances lead to different perturbation kernels and hence different stability thresholds. A local version gives local uniqueness and quantitative attraction near a prescribed equilibrium. We apply the framework to two representative models. For non-symmetric granular media dynamics, we derive two explicit uniqueness thresholds that reveal the topology-sensitive nature of the theory. For the dynamical Curie–Weiss model, the Wasserstein-1 criterion recovers the sharp bifurcation threshold up to the critical equality. We identify all invariant measures, establish basin-dependent exponential convergence in the phase-transition regime, and show that the loss of anchored smallness at criticality leads to polynomial slowing down.

We extend the spherical-integral approach to $\mathcal{R}$-transforms of non-Hermitian random matrices to random matrices with a fixed $K \times K$ block structure. We introduce a rank-$K$ spherical integral defining a scalar $H$-transform on $2K \times 2K$ overlap matrices and an associated matrix-valued $\mathcal{R}$-transform encoding the joint block structure. Using the replica method, we derive a conjectural subordination relation for sums of independent block matrices. This framework provides a method for determining the spectral boundaries of large non-Hermitian random matrices with a fixed block structure.

We establish a cutoff criterion for finite Markov chains with non-negative Bakry–Émery curvature without assuming that the support of the transition matrix is symmetric. This extends a criterion of Salez based on a refined product condition. For each allowed transition, we consider a shortest directed path from its endpoint back to its starting point. Our cutoff criterion depends on the maximum length of these paths. We also obtain a discrete-time counterpart for lazy chains. For random walks on finite Abelian groups, we prove non-negative Bakry–Émery curvature for arbitrary generating sets. As an application, we establish almost-sure cutoff for random walks on products of cyclic groups of order three.

The stochastic block model is a paradigmatic model of networks with community structure. Yet percolation in the model has been studied primarily in cases with a fixed block structure, even though in real networks, the community structure may evolve as the network grows. Here we study percolation in sequences of stochastic block models in which the numbers and sizes of communities, as well as the intra- and intercommunity connection probabilities may all change with the network size. We analyze five such sequences using two methods: linearized self-consistent equations for the locally tree-like models and a branching process at the community scale for the models with nonvanishing clustering. We find that the critical average degree is not generally equal to $1$, even in locally tree-like sequences, because the transition depends on how connections are distributed across the evolving community structure. We also show that the community-scale branching process accurately predicts the transition when intercommunity connections are sufficiently sparse, even in the presence of nonvanishing clustering, while a geometric stochastic block model sequence demonstrates the limitations of this method when correlations between intercommunity connections cannot be neglected. These results extend percolation studies in the stochastic block model to more realistic scenarios with evolving community structure, and may provide new methods to derive the upper and lower bounds for the percolation threshold in geometric long-range percolation.

We obtain a complete characterization of the quasi-stationary distributions of subcritical stable continuous-state branching processes in Brownian random environments, together with the domain of attraction of each member. These distributions form a one-parameter family whose modified Laplace transforms are given explicitly in terms of Tricomi's confluent hypergeometric function. For every initial distribution, we determine whether the conditional law converges to a QSD and identify that QSD whenever it does. The necessary and sufficient criteria are formulated in terms of the tail probabilities and moments of the initial law.

We study equilibrium and risk evaluation in large sparse economic networks with heterogeneous responses, shocks, and bilateral exposures. Our approach approximates the distribution of equilibrium outcomes through local computations on a limiting rooted network. Under marked local weak convergence, we prove convergence in probability of the empirical equilibrium distribution in two settings: uniformly contractive responses and bounded monotone nonexpansive responses whose lower and upper root iterations coalesce. In the latter setting, the limit is independent of the measurable finite-network equilibrium selection, even when equilibria are not unique. Under bounded domain and continuity assumptions, this convergence extends to law-invariant risk measures, including conditional value-at-risk, with recursion-depth error bounds under additional regularity. Numerical experiments on production networks and payment clearing systems illustrate the accuracy and computational benefits of the local approximation.

We consider the dynamic Schrödinger bridge problem on a finite state space and exploit its Markov structure to decompose the problem into an endpoint coupling and a collection of conditional Markov bridges. To sample from the conditional bridges, we develop a quantum Gibbs sampler based on quantum walks on the bridge path space. Under uniformly positive transition probabilities, we derive a spectral-gap bound and provide an explicit procedure for preparing the initial state. Compared with the update bounds of the corresponding classical Gibbs sampler, our quantum walk-query complexity improves the dependence on the time horizon $T$ from quadratic to linear. For the endpoint coupling, we adapt a quantum box-constrained Newton method to compute the Schrödinger potentials, improving the matrix-scaling complexity, at fixed accuracy, from quadratic to $N^{3/2}$ in the state-space size $N$. Finally, we show that exponential reweighting extends both the Schrödinger bridge formulation and the Gibbs spectral-gap estimate to models with additive running costs.

Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of $\nu$ are the lower and upper limits of $\log(1/t)/\mathcal{L}_{\nu}(t)$. The dimension exists precisely when $(q-1)\sum_{Q}\nu(Q)^{q}$, summed over all dyadic cubes, converges as $q\downarrow1$, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width $1/k$ around $q=1$; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in $(0,1]$, and a regime-switching example separates convergence in law from almost-everywhere convergence.

We find a new coupling between the Airy sheet $\mathcal{S}$ and the Airy line ensemble $\{ \mathcal{A}_n \}_{n \in \mathbb{N}}$, which we call the down-the-middle isometry. In this coupling, the Airy sheet is encoded by last passage values that start at $\mathcal{A}_k(0)$ for some $k \in \mathbb{N}$ and end on the top line $\mathcal{A}_1$. Using this coupling, we give a new construction of the (extended) Airy sheet as a scaling limit of Brownian last passage percolation and study coalescence of geodesics in the directed landscape. This coupling extends to discrete polymers and last passage percolation models, as well as to the KPZ line ensemble and KPZ sheet.

The Odlyzko-Poonen conjecture states that a monic polynomial of constant coefficient $1$ and with remaining coefficients chosen independently and uniformly from $\{0,1 \}$ is irreducible over the rational numbers with probability tending to one as the degree tends to infinity. We prove the Odlyzko-Poonen conjecture unconditionally.

For every $n\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\ge3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\ge4$. This number is optimal when $r=2^k+2$, $k\ge1$.

In sparse random matrices, spectral outliers (eigenvalues and singular values located away from the bulk) emerge due to degree fluctuations: high degrees inflate the operator norm, while low column degrees reduce the least singular value. As proved by Feige and Ofek (2005) and Le, Levina, and Vershynin (2017), degree regularization enforces concentration at the expected norm scale. However, precise bounds incorporating the cutoffs remain unexplored and challenging since regularization introduces dependencies among entries. For the first time in the literature, we provide variance- and cutoff-dependent bounds for extreme singular values and eigenvalues of regularized inhomogeneous random matrices. In the absence of regularization, our lower bound for the least singular value matches the same leading constant obtained by Brailovskaya and van Handel (2024). Moreover, our error term vanishes under the milder condition $d/\log N\to\infty$, as opposed to their stronger requirement $d/(\log N)^4\to\infty$. A key ingredient is to extend spectral radius bounds for nonbacktracking matrices to the dependent setting. We build on approaches for independent cases established by Benaych-Georges, Bordenave, and Knowles (2020), as well as Dumitriu and Zhu (2024), and carefully handle edges traversed only once. Our proof framework separates deterministic spectral comparisons from probabilistic estimates: once Loewner inequalities and columnwise variance controls are established, the remaining probabilistic analysis boils down to verifying the graph moment conditions formulated in this paper. We hope this framework can be extended to handle general random matrices with more complex dependencies.

We study scaling limits of $\mathbb R^d$-valued branching random walks (BRWs for short) in the following setting: indexing-trees are Galton-Watson trees with critical offspring distribution which are assumed, when appropriately rescaled, to converge to a critical $\psi$-Lévy tree; conditional on the indexing tree, the jumps of the BRW may exhibit dependence within the same sibling group, but distinct sibling groups are independent, and their distributions are centered but may vary (and for instance depend on the whole indexing tree), with a typical value of the walk nevertheless remaining within the domain of attraction of a mixture of Gaussian distributions. Under these assumptions and a necessary additional assumption, which we call the discrete Sheu assumption, we show that rescaled discrete snakes converge functionally to the $\psi$-Brownian snake introduced by Le Gall & Le Jan (1998) and D.~& Le Gall (2002). The method uses a coupling result for BRWs of independent interest. An application is given concerning the scaling limits of the range of BRWs that take their values in the $\mathtt b$-ary tree to the reflected $\psi$-Brownian cactus which is a variant of the Brownian cactus introduced by Curien, Le Gall & Miermont (2013).

We establish sharp variance asymptotics and limiting distributions for the quadratic optimal matching cost between the uniform measure and an empirical measure counterpart on the flat torus $\mathbb T^d$ in dimensions $d=2,3,4$. More precisely, let $\mu$ be Haar probability measure and $\mu_n$ the empirical measure of $n$ independent samples drawn from $\mu$. Then, for $d=2,3$, $$ \begin{aligned} n\bigl(W_2^2(\mu_n,\mu)-\mathbb E[W_2^2(\mu_n,\mu)]\bigr) &\xrightarrow{\mathrm d} L_d,\\ n^2\operatorname{Var}(W_2^2(\mu_n,\mu)) &\longrightarrow \frac{1}{8\pi^4} \sum_{k\in\mathbb Z^d\setminus\{0\}}\frac{1}{|k|^4}, \end{aligned} $$ where $L_d$ is an explicit non-Gaussian weighted sum of independent centered exponential random variables. In dimension four, $$ \frac{n}{\sqrt{\log n}} \bigl(W_2^2(\mu_n,\mu)-\mathbb E[W_2^2(\mu_n,\mu)]\bigr) \xrightarrow{\mathrm d} N\left(0,\frac{1}{16\pi^2}\right). $$ The main idea of the proof is to rely on a Hoeffding decomposition of the optimal transport cost, which turns out to be asymptotically equivalent to U-statistics of order 2.

We study a passive scalar on the two-dimensional torus driven by an arbitrary finite set of white-in-time divergence-free Fourier modes. We prove that the exponential $L^2$ dissipation rate of the advection-diffusion equation is uniformly bounded as the diffusivity tends to zero as long as there exist two nonparallel forced modes. If not, then the dissipation rate can be arbitrarily fast depending on the initial data, but the top Lyapunov exponent remains uniformly bounded below. The proof uses a projection onto a finite set of modes which depends on the forcing set. Part of our argument follows the structure of the four-mode case investigated by Chemnitz and Chemnitz, but uses a strictly weaker condition than their matrix inequality between Frobenius and operator norms. Combined with an accessibility argument, this allows us to consider an arbitrary set of forcing modes and initial data.

In 2015, Bondesson showed that Thorin's class of generalized gamma convolutions (GGCs) has the remarkable property of being closed under multiplication of independent random variables. He also conjectured that the GGC class is closed under taking powers of order greater than one. In this paper, we provide a candidate proof for Bondesson's conjecture.

We study the high-temperature derivation of $\Phi^4_d$ measures from grand-canonical Bose gases on two- and three-dimensional tori. Building on the upper-symbol variational approach of the author's earlier work, we use relative entropy to give a simple, unified proof of relative free-energy convergence and Hilbert–Schmidt convergence of all fixed-order rescaled reduced density matrices to their Hartree counterparts. For interaction ranges $\varepsilon=\lambda^\eta$, where $\lambda$ is the inverse temperature, these results hold for every fixed $0<\eta<1/2$ in two dimensions and $0<\eta<1/26$ in three. In two dimensions, we improve the result of Jougla and Rougerie (2026) and the local $\Phi^4_2$ derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local $\Phi^4_2$ derivation valid for every fixed exponent $0<\eta<1/2$, allowing polynomial exponents arbitrarily close to the diluteness threshold from below. In three dimensions, we use the identification of the local measure and the stationary SPDE estimates of Nam, R.~Zhu and X.~Zhu (2025). From these estimates we strengthen their classical correlation convergence to the Hilbert–Schmidt norm. This yields the local limit at every fixed order in the same polynomial range.

In classical general relativity, it is well known that radially moving massless particles in the exterior region take infinite coordinate time to reach the event horizon in black hole spacetimes. In this short paper, we use two stochastic versions of ingoing radial null geodesic equation to study the stochastic radial motion of massless particles in Schwarzschild and Reissner-Nordström spacetimes. We show that the probability that a radially moving massless particle enters the black hole in finite coordinate time remains zero. This indicates the stochastic perturbations of quantum effects preserve certain properties of classical black holes.

The KPZ upper-tail field introduced by Liu and Zhang arises as the scaling limit of the narrow-wedge KPZ fixed point in a microscopic neighborhood of a conditioned large height. In this paper, we give a direct probabilistic identification of the upper-tail field. We identify the grounded positive-time sector with the KPZ fixed point started from the two-sided Brownian \(V\)-profile \(B_{\mathrm{ts}}(2x)-2|x|\), while the negative-time sector is represented through an explicit exponential tilt of the same evolution. The argument uses the directed-landscape variational structure and the local Brownian comparison under upper-tail conditioning. As consequences of these representations, we recover probabilistically the positive- and negative-time large-scale limits of the KPZ upper-tail field.

The concept of weighted universal Value-at-Risk superadditivity (WUVS) was recently introduced by Chen et al. (2026) as a generalization of the question whether for some infinite mean distributions convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. In this short note we prove that the property WUVS can basically never hold for discrete distributions except for the case of comonotonicity. This implies as a corollary that for discrete distributions with infinite mean it can also never hold that convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. This settles an open problem mentioned in Müller (2025).

These lecture notes are intended as an introduction to the theory of rational Dunkl operators, the associated special functions and related Markov processes with an emphasis on examples which are related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones of positive semidefinite matrices. We start with a comprehensive introduction into Dunkl theory: Dunkl operators, the intertwining operator and its positivity, the Dunkl kernel and the Dunkl transform, the Dunkl Laplacian and the associated heat semigroup. We further give an outline of the connection with Calogero-Moser-Sutherland models and generalized Hermite polynomials. Moreover, of central interest will be product formulas, generalized translations and associated commutative hypergroup structures on closed Weyl chambers. In particular, we explain how Dunkl theory for particular multiplicities is related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones, and how this leads to a bunch of multiplicities for which the Weyl-group invariant Dunkl theory admits a probability preserving translation and an associated commutative hypergroup structure on the closed Weyl chamber. We finally discuss Markov processes on R^N which are related with Dunkl theory with an emphasis on connections to random walk on groups and hypergroups. In particular associated martingales, martingale characterizations, moment functions and Appell characters are studied, i.e. diffusion-reflection processes with the Dunkl Laplacians as generators.

We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For $\beta=2$, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal conditions. This extends the seminal results of Katori and Tanemura. The additional parameters record information at infinity and enter through an associated Laguerre-Pólya entire function. Moreover, for explicit classes of configurations, we establish convergence on path space and the Markov property. We prove rescaled long-time convergence, in finite-dimensional distributions, to the stationary extended $\mathsf{Sine}$ process from arbitrary symmetric initial configurations with power-law counting exponent $q\in(0,2)$. This extends the integer lattice relaxation result of Katori and Tanemura which was the only such result for explicit deterministic initial conditions. For $\beta\geq1$, we prove convergence of finite particle systems from regular initial data to the unique strong solution of an infinite-dimensional stochastic differential equation in a certain rigid-path-regularity class. This extends seminal works of Tsai and Osada. We finally derive a stochastic partial differential equation of Burgers-type for the Stieltjes transform of the dynamics.

This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power $p$ converges to a dimension-independent constant multiple of the corresponding $L^p$ norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian–Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.

We study the behavior at infinity of the invariant Radon measure for the multidimensional affine stochastic recursion $V_n = A_n V_{n-1} + B_n,$ where $(A_n)_{n \geq 1}$ are positive random matrices, $(B_n)_{n \geq 1}$ are random vectors with nonnegative entries, and $(A_n, B_n)_{n \geq 1}$ are independent and identically distributed. In the critical regime where the top Lyapunov exponent of the random matrix products $A_n \cdots A_1$ is zero, Brofferio, Peigné and Pham [6] recently established the existence and uniqueness, up to multiplication by a constant, of an invariant Radon measure with infinite total mass. They proved that the tail behavior of this measure when applied to radial sets is governed by a slowly varying function. Our goal is to show that this slowly varying function is actually bounded. Moreover, we investigate directional tail behavior.

We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent $\alpha\in (0,1)$; our examples include the Euler, SQG and IPM systems. We investigate whether a number of “turbulent” phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of $\alpha$ and initial data in $L^p_x$, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity $\beta=\beta(\alpha,p)>0$. Remarkably, these results allow for some scaling supercritical regimes of the parameters $\alpha,p$; on the other hand, for (sub)critical parameters, we recover the same regularity exponent $\beta=1-\alpha$ as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously $L^\infty_x$-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for $\alpha\in (0,1/2)$, we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.

Cancer progression often requires multiple driver mutations, but the same drivers may be acquired in different orders. How these pathways jointly shape tumor clonal structure remains unclear. We develop a multitype branching-process model in which malignant transformation requires two driver mutations, distinguishing malignant cells by mutation order and the independent transformation event that founded their clone. Under a successive exponential approximation, we establish point-process limits for pathway-specific clone sizes and derive a closed-form expression for the limiting expected Simpson's index of the combined malignant population. When both mutation orders yield malignant cells with the same net growth rate, the index decomposes into effective pathway weights, determined by mutation rates and birth-death dynamics at preceding stages, and within-pathway concentration terms, determined by intermediate-to-malignant growth-rate ratios. A driver's effect on heterogeneity thus depends critically on when it is acquired. A strong driver acquired early expands the intermediate lineage and increases the supply of independent malignant founders, whereas the same driver acquired last strengthens the growth and age advantage of early-founded malignant clones. Under additive fitness effects, these opposing mechanisms can produce a non-monotone relationship between selective advantage and clonal concentration. Threshold-like non-additive fitness effects can generate highly concentrated malignant populations, while order-dependent terminal fitness causes the faster-growing pathway to dominate asymptotically. These results show how mutation order, mutational accessibility, selection, and epistasis jointly determine lineage-level intratumor heterogeneity.

A functional central limit theorem for locally stationary time series taking values in a separable Banach space $B$ is established. The result does not require type-2 or cotype assumptions and therefore covers spaces central to functional data analysis, including $C([0,1])$ and $L^p([0,1])$. Under moment conditions, summable physical dependence coefficients, and a bracketing entropy condition controlling the infinite-dimensional tails, the centered and rescaled partial sum process converges weakly in $D([0,1],B)$. The limit is a centered $B$-valued Gaussian process whose covariance is given by the integral of the local long-run covariance, interpreted as an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion, connecting the Banach-space limit to the familiar scalar locally stationary structure. As an application, we derive a self-normalized CUSUM procedure for detecting changes in the mean of linear projections of Banach-valued observations, yielding a pivotal asymptotic null distribution. The finite-sample behavior is illustrated through Monte Carlo experiments and exploratory applications to EEG recordings and daily temperature curves. Examples based on the Faber-Schauder system in $C([0,1])$ and on a $p$-Laplacian model in $W^{1,p}_0([0,1])$ demonstrate how the entropy condition can be verified.

This paper is concerned with stochastic Hamiltonian systems driven by Lévy processes. By employing the lent particle method developed by Bouleau and Denis in the framework of Malliavin calculus with jumps, we establish an explicit Bismut–Elworthy–Li type derivative formula for the associated Markov semigroup, as well as corresponding gradient estimates. The main novelty lies in extending the known results for Brownian-motion-driven degenerate systems to the nonlocal setting with jumps, where the presence of a pure jump noise and the degeneracy of the coefficients pose substantial difficulties. Our approach provides a systematic treatment of such nonlocal degenerate operators and fills a gap in the literature on derivative formulae for jump-type stochastic Hamiltonian systems.

We consider the superposition $S_n$ of $n$ i.i.d. simple point processes on $\mathbb{R}^d$ and apply a dependent thinning $T_n$, where the retention probability of a point depends on its local neighbour count within a radius $r_n$. While superpositions under independent thinning converge to Poisson processes, we show that under a critical geometric scaling $n v_d r_n^d \to \tau \in (0,\infty)$, the local interactions are transformed in the limit into an inhomogeneous Poisson point process with modified intensity. We prove that the thinned sequence $T_n(S_n)$ converges to a Poisson process with a non-linearly modified intensity $\tilde\lambda(x) = \lambda(x)\alpha(\tau\lambda(x))$.

Repeated averaging on graphs is a natural prototype of local stochastic exchange, modeling consensus formation, information spreading, and wealth redistribution. Its convergence has attracted longstanding interest, from Bourgain's complete-graph $L^1$ cutoff question in the 80s (answered in recent years by Chatterjee, Diaconis, Sly, and Zhang) and the systematic study of Aldous and Lanoue. Addressing the still outstanding and widely asked $L^1$-cutoff question on general graphs, we establish a criterion based on size-biased sampling, and develop a general framework for sparse expanders. Specifically, we prove $L^1$ cutoff with a Gaussian profile on random $d$-regular graphs, for any $d\ge 3$ and uniformly over starting vertices. Remarkably, although the expected mass profile agrees exactly with the corresponding random-walk distribution, cutoff occurs strictly later than both the random walk cutoff time and the universal lower bound attained on complete graphs, revealing an unexpected entropy gap. The framework is robust, involving ideas of polymer representations, random geometry, universal-cover lifting, and adaptive revealing. Our companion work will extend it to Ramanujan graphs and bounded-degree configuration models.