1 Introduction↩︎

The broad aim of the present work is to investigate the nature of the transition from out of equilibrium to equilibrium in MCMC algorithms, a popular class of methods for sampling from a target measure \(\pi\in\mathcal{P}\tond*{\mathbb{R}^d}\) by simulating a stochastic process that is ergodic with respect to \(\pi\). We will here focus on two particular implementations that are widely used in practice: the Langevin dynamics, and the Proximal Sampler. Throughout the paper, we will assume that the target measure \(\pi\) is log-concave, i.e. of the form \[\begin{align} \pi({\mathrm d}x) & = & e^{-V(x)}{\mathrm d}x, \end{align}\] for some convex potential \(V\in C^2(\mathbb{R}^d)\).

1.1 The Langevin dynamics↩︎

The Langevin dynamics with target distribution \(\pi\) and initialization \(\mu_0\in \mathcal{P}\tond*{\mathbb{R}^d}\) is simply the solution to the stochastic differential equation \[\begin{align} \label{eq:LD} X_0\sim \mu_0, \quad {\mathrm d}X_t & = & -\nabla V(X_t)\,{\mathrm d}t + \sqrt{2}\,{\mathrm d}B_t, \end{align}\tag{1}\] where \((B_t)_{t\ge 0}\) is a standard \(d-\)dimensional Brownian motion. Under our assumptions, it is well known that the marginal law \(\mu_t\mathrel{\vcenter{:}}=\mathop{\mathrm{law}}(X_t)\) approaches \(\pi\) as \(t\to\infty\). Moreover, a variety of quantitative convergence guarantees are available in different metrics; see, e.g., bak-gen-led-2014? for an extensive treatment. In particular, the parabolic regularization estimate \[\begin{align} \label{eq:KL-W2-reg} \forall t>0,\quad {H}\tond*{\mu_t\,\middle |\, \pi} & \leq & \frac{W^2(\mu_0,\pi)}{4t}, \end{align}\tag{2}\] was proved in bob-gen-led-2001?, ott-vil-2001?, and used to recover and generalize the celebrated HWI inequality from ott-vil-2000?. Here and throughout the paper, we use the classical notation \[\begin{align} W^2(\mu,\pi) & \mathrel{\vcenter{:}}= & \inf_{X\sim\mu, Y\sim \pi} \mathbb{E}\quadr*{\abs*{X-Y}^2}, \end{align}\] for the \(2\)-Wasserstein distance between \(\mu\) and \(\pi\), and \[\begin{align} {H}\tond*{\mu\,\middle |\, \pi}& \mathrel{\vcenter{:}}= & \int \log\tond*{\frac{\mathrm{d}\mu}{\mathrm{d}\pi}} \mathrm{d}\mu, \end{align}\] for the relative entropy (or KL-divergence) of \(\mu\) with respect to \(\pi\). As usual, it is understood that \({H}\tond*{\mu\,\middle |\, \pi}=\infty\) when \(\mu\) is not absolutely continuous with respect to \(\pi\). To translate 2 into the classical language of mixing times, we let \[\begin{align} \mathrm{TV}\tond*{\mu, \pi} & \mathrel{\vcenter{:}}= & \sup_{A\in \mathcal{B}(\mathbb{R}^d)}\abs*{\mu(A)-\pi(A)} \end{align}\] denote the total-variation distance between \(\mu\) and \(\pi\), and we recall that the mixing time of the process \((X_t)_{t\ge 0}\) with initialization \(\mu_0\) and precision \(\varepsilon\in(0,1)\) is defined as \[\begin{align} \label{def:mixing} \mathrm{t}_{\mathrm{mix}}(\mu_0,\varepsilon) & \mathrel{\vcenter{:}}= & \min\{t\ge 0\colon \mathrm{TV}\tond*{\mu_t, \pi}\le\varepsilon \}. \end{align}\tag{3}\] It then readily follows from 2 and Pinsker’s inequality that \[\begin{align} \label{eq:tmixLS} \forall \varepsilon\in(0,1),\quad \mathrm{t}_{\mathrm{mix}}(\mu_0,\varepsilon) & \le & \frac{W^2(\mu_0,\pi)}{8\varepsilon^2}. \end{align}\tag{4}\] In concrete words, running the Langevin dynamics for a time of order \(W^2(\mu_0,\pi)\) suffices to approximately sample from \(\pi\). Unfortunately, the continuous-time nature of the Langevin dynamics is not appropriate for practical implementation, and a suitable discretization is required. The simplest choice is the Euler–Maruyama method, in which a step size \(h>0\) is chosen and a discrete-time process \((X_k)_{k\in\mathbb{N}}\) is produced via the stochastic recursion \[\begin{align} \label{eq:LMC} X_0\sim\mu_0,\quad X_{k+1} & = & X_k -h\nabla V(X_k) + \sqrt{2}\tond*{B_{(k+1)h} - B_{kh}}. \end{align}\tag{5}\] This sampling scheme is known as the Langevin Monte Carlo (LMC) algorithm or Unadjusted Langevin algorithm. It is of fundamental importance, and has been extensively studied. We refer the unfamiliar reader to the book in progress che-book-2024+? and the references therein. One drawback of this approach, however, is that the LMC algorithm is biased: the stationary distribution of 5 is in general different from \(\pi\). As a consequence, theoretical performance guarantees require the step size to be very small, resulting in an increased number of iterations compared to the theoretical time-scale 4 .

1.2 The Proximal Sampler↩︎

The Proximal Sampler is an unbiased discrete-time algorithm for sampling from \(\pi\) introduced in lee-she-tia-2021?. We refer the reader to che-book-2024+? or the recent papers che-eld-2022?, mit-wib-2025?, wib-2025? for more details. As usual, we denote by \(\gamma_{x,t}\) the \(d\)-dimensional Gaussian law with mean \(x\) and covariance matrix \(tI_d\), and we use the shorthand \(\gamma_t \mathrel{\vcenter{:}}= \gamma_{0,t}\). Given a step size \(h>0\), we consider a pair \((X,Y)\) of \(\mathbb{R}^d-\)valued random variables with joint law \[\begin{align} \boldsymbol{\pi}({\mathrm d}x\, {\mathrm d}y) & \propto & \exp\tond*{-V(x) - \frac{\abs*{x-y}^2}{2h}}{\mathrm d}x\, {\mathrm d}y. \end{align}\] The Proximal Sampler with target \(\pi\) consists in applying alternating Gibbs sampling to \(\boldsymbol{\pi}\). More precisely, a sequence \(X_0,Y_0,X_1,Y_1,\ldots\) of \(\mathbb{R}^d\)-valued random variables is produced by first sampling \(X_0\) according to some prescribed initialization \(\mu_0\in \mathcal{P}\tond*{\mathbb{R}^d}\) and then inductively, for each \(k\ge 0\), sampling \(Y_k\) and \(X_{k+1}\) according to the following laws:

1. Forward step: conditionally on \((X_0,Y_0,\ldots,X_{k-1},Y_{k-1},X_k)\), \(Y_k\) is distributed as \[\begin{align} \label{eq:forward-step-prox} Y_k & \sim & \mathop{\mathrm{law}}\tond*{ Y \mid X = X_k} \;= \;\gamma_{X_k, h}. \end{align}\tag{6}\]

2. Backward step: conditionally on \((X_0,Y_0,\ldots,X_k,Y_k)\), \(X_{k+1}\) is distributed as \[\begin{align} \label{eq:backward-step-prox} X_{k+1} & \sim & \mathop{\mathrm{law}}\tond*{ X \mid Y = Y_k} \;\propto \;\exp \tond*{-V(x)-\frac{\abs*{x-Y_k}^2}{2h}}\mathrm{d}x. \end{align}\tag{7}\]

Since the first marginal of \(\boldsymbol{\pi}\) is \(\pi\), the algorithm is unbiased, meaning that \(\pi\) is stationary for the Markov chain \((X_k)_{k\ge 0}\). Moreover, the forward step is trivial to implement, as it amounts to adding an independent Gaussian noise. When \(h\) is small enough, the backward step is also tractable, due to the regularizing effect of the quadratic potential in 7 . For example, if \(\pi\) is \(\alpha\)-log-concave and \(\beta\)-log-smooth (i.e. \(\alpha I_d\preccurlyeq \nabla^2V \preccurlyeq \beta I_d\)), then the conditional law 7 is \(\tond*{\alpha + \frac{1}{h}}\)-log-concave with condition number \(\kappa_h = \frac{1+\beta h}{1+\alpha h} < \frac{\beta}{\alpha}\). As a result, several methods are available to efficiently generate \(X_{k+1}\), such as rejection sampling che-che-sal-wib-2022?, approximate rejection sampling fan-yua-che-2023?, or high-accuracy samplers alt-che-2024-faster?. Following a standard convention in this setting che-che-sal-wib-2022?, we will here assume to have access to a restricted Gaussian oracle that implements the backward step 7 exactly, and focus on the number of iterations needed for \(\mu_k\mathrel{\vcenter{:}}=\mathop{\mathrm{law}}(X_k)\) to approach \(\pi\), in the sense of 3 .

Although this is not obvious from the above description, the Proximal Sampler can be interpreted as a discretization of the Langevin dynamics 1 . This is particularly clear once we view those dynamics as minimizing schemes for the relative entropy functional \({H}\tond*{\cdot\,\middle |\, \pi}\) in the Wasserstein space \(\tond*{\mathcal{P}_2(\mathbb{R}^d), W}\) jor-kin-ott-1998?, che-book-2024+?. In light of this, it is not too surprising that many classical convergence guarantees for the Langevin dynamics translate to the Proximal Sampler. In particular, the following analogue of the parabolic regularization estimate 2 was recently established in che-che-sal-wib-2022?: \[\begin{align} \label{eq:KL-W2-reg-prox-sampl} \forall k\in\mathbb{N},\quad {H}\tond*{\mu_k\,\middle |\, \pi} & \leq & \frac{W^2\tond*{\mu_0,\pi}}{kh}. \end{align}\tag{8}\] By virtue of Pinsker’s inequality, this readily yields the (discrete-time) mixing-time estimate \[\begin{align} \forall \varepsilon\in(0,1),\quad \mathrm{t}_{\mathrm{mix}}(\mu_0,\varepsilon) & \le & \left\lceil\frac{W^2(\mu_0,\pi)}{2h\varepsilon^2}\right\rceil. \end{align}\] In concrete words, running the Proximal Sampler with step size \(h\) for roughly \(W^2\tond*{\mu_0,\pi}/h\) iterations suffices to produce approximate samples from the target distribution \(\pi\).

1.3 Main results↩︎

Rather than asking how long the Langevin dynamics or the Proximal Sampler should be run in order to be close to equilibrium, we would here like to understand how abrupt their transition from out of equilibrium to equilibrium is. In other words, we seek to estimate the width of the mixing window, defined for any precision \(\varepsilon\in\tond*{0,\frac{1}{2}}\) by \[\begin{align} \mathop{\mathrm{w_{mix}}}(\mu_0,\varepsilon) & \mathrel{\vcenter{:}}= & \mathrm{t}_{\mathrm{mix}}(\mu_0,\varepsilon)-\mathrm{t}_{\mathrm{mix}}(\mu_0,1-\varepsilon). \end{align}\] The analysis of this fundamental quantity is a challenging task which, until recently, had only been carried out in a handful of models. Over the past couple of years, a systematic approach to this problem was developed in the series of works sal-2023?, sal-2024?, sal-2025?, ped-sal-2025?, using an information-theoretic notion called varentropy. In the present paper, we propose an alternative approach which completely bypasses the use of varentropy, and instead exploits the parabolic regularization estimates 2 and 8 directly, in conjunction with a new W-TV transport inequality. As a first application, we are able to recover exactly the main result of sal-2025?, which reads as follows. Recall that the Poincaré constant of \(\pi\in\mathcal{P}\tond*{\mathbb{R}^d}\), denoted \(C_{\rm{P}}\tond*{\pi}\), is the smallest number such that \[\begin{align} \label{def:PI} \mathrm{Var}_{\pi}\tond*{f} & \leq & C_{\rm{P}}\tond*{\pi} \int \abs*{\nabla f}^2 d\pi, \end{align}\tag{9}\] for all smooth functions \(f\colon \mathbb{R}^d\to\mathbb{R}\). In particular, \(C_{\rm{P}}\tond*{\delta_x}=0\), for any \(x\in\mathbb{R}^d\).

The interest of this estimate lies in its relation to the cutoff phenomenon, a remarkably universal – but still largely unexplained – phase transition from out of equilibrium to equilibrium undergone by certain ergodic Markov processes in an appropriate limit. We refer the unfamiliar reader to the recent lecture notes salez2025modernaspectsmarkovchains? and the references therein for an up-to-date introduction to this fascinating question.

Note that, in the standard setup where the initialization is a Dirac mass, our cutoff criterion 10 reduces to the natural condition \(\frac{\mathrm{t}_{\mathrm{mix}}(\mu_0,\varepsilon)}{C_{\rm{P}}\tond*{\pi}}\to\infty\), which is known as the product condition in the classical literature on Markov chains (see, e.g., salez2025modernaspectsmarkovchains?).

As a second – and genuinely new – application of our transport approach to cutoff, we extend the above results to the Proximal Sampler, thereby tightening its relation to the Langevin dynamics. To lighten the formulas, we introduce the quantity \[\begin{align} \widehat{C}_{\rm{P}}\tond*{\pi} & \mathrel{\vcenter{:}}= & 1+\frac{C_{\rm{P}}\tond*{\pi}}{h}, \end{align}\] which, as we will see, can be seen as the natural discrete-time analogue of \(C_{\rm{P}}\tond*{\pi}\).

Here again, the condition 10 reduces to \(\frac{\mathrm{t}_{\mathrm{mix}}(\mu_0, \varepsilon)}{\widehat{C}_{\rm{P}}\tond*{\pi}}\to\infty\) for Dirac initializations. To the best of our knowledge, this is the very first result establishing cutoff for the Proximal Sampler. We emphasize that the latter is a discrete-time Markov process on a continuous state space, an object to which the varentropy approach developed in sal-2023?, sal-2024?, sal-2025?, ped-sal-2025?, salez2025modernaspectsmarkovchains? does not currently apply. Indeed, in that series of work, varentropy is controlled using either the celebrated chain rule, which notoriously fails in discrete time, or an approximate version of it involving a certain sparsity parameter, which only makes sense on discrete spaces. To bypass this limitation, our main idea is to replace the reverse Pinsker inequality sal-2023? where varentropy appears with the following W-TV transport inequality, which seems new and of independent interest.

Acknowledgment↩︎

F.P. thanks Yuansi Chen for helpful comments. J.S. is supported by the ERC consolidator grant CUTOFF (101123174). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible.

2 Proofs↩︎

2.1 The W-TV transport inequality↩︎

In this section, we prove Theorem 4. Given two probability measures \(\mu,\nu \in \mathcal{P}\tond*{\mathbb{R}^d}\), we recall that the chi-squared divergence of \(\nu\) w.r.t. \(\mu\) is defined by the formula \[\begin{align} \chi^2\tond*{\nu\,\middle |\, \mu} & \mathrel{\vcenter{:}}= & \int \tond*{\frac{\mathrm{d}\nu}{\mathrm{d}\mu}-1} \mathrm{d}\nu, \end{align}\] with \(\chi^2\tond*{\nu\,\middle |\, \mu}=\infty\) if \(\nu\) is not absolutely continuous w.r.t. \(\mu\). Our starting point is the following transport-variance inequality, whose proof can be found in liu-2020?.

Unfortunately, the chi-squared divergence appearing here could be arbitrarily large compared to the total-variation term with which we seek to control \(W^2(\mu,\nu)\). To preclude such pathologies, we introduce a probability measure \(\lambda\) which interpolates nicely between \(\mu\) and \(\nu\), in the sense of having small Radon-Nikodym derivatives w.r.t. both.

We now have everything we need to establish Theorem 4.

2.2 Cutoff for the Langevin dynamics↩︎

In this section, we prove Theorem 1 and Corollary 1. Consider the Langevin dynamics 1 with target \(\pi\) and initialization \(\mu_0\in\mathcal{P}\tond*{\mathbb{R}^d}\), and write \(\mu_t = \mathop{\mathrm{law}}(X_t)\) for the law at time \(t\ge 0\). As is well known from the Bakry–Émery theory bak-gen-led-2014? (see Remark 5 below for an alternative proof), the log-concavity of \(\pi\) ensures the local Poincaré inequality \[\begin{align} \label{eq:LPI} \forall t\ge 0,\quad C_{\rm{P}}\tond*{\mu_t} & \le & C_{\rm{P}}\tond*{\mu_0}+2t. \end{align}\tag{13}\] Another ingredient that we will need is the basic mixing-time estimate \[\begin{align} \label{eq:sal-fast-tv-mixing-small-kl} \mathrm{t}_{\mathrm{mix}}\tond*{\mu_0,\varepsilon} & \leq & \frac{C_{\rm{P}}\tond*{\pi}\tond*{1+{H}\tond*{\mu_0\,\middle |\, \pi}}}{\varepsilon}, \end{align}\tag{14}\] borrowed from sal-2023?, and which relies on the classical fact that \[\begin{align} \forall t\ge 0,\quad \chi^2\tond*{\mu_t\,\middle |\, \pi} & \le & e^{-2C_{\rm{P}}\tond*{\pi}t} \chi^2\tond*{\mu_0\,\middle |\, \pi}, \end{align}\] together with an easy interpolation argument between \(\chi^2\tond*{\mu_t\,\middle |\, \pi},{H}\tond*{\mu_t\,\middle |\, \pi}\) and \(\mathrm{TV}\tond*{\mu_t, \pi}\).

2.3 Cutoff for the Proximal Sampler↩︎

To prove Theorem 3, we fix a log-concave target \(\pi\in\mathcal{P}\tond*{\mathbb{R}^d}\), a step size \(h>0\), and an initialization \(\mu_0\in\mathcal{P}\tond*{\mathbb{R}^d}\), and we consider the random sequence \(X_0,Y_0,X_1,Y_1,\ldots\) generated by the Proximal Sampler 6 7 . We write \(\mu_k=\mathrm{law}(X_k)\). The main ingredient we need in order to mimic the proof of Theorem 1 is a version of the local Poincaré inequality 13 for the Proximal Sampler, provided in the following lemma.

We will also need the following analogue of the mixing-time estimate 14 .

We now have everything we need to mimic the proof of Theorem 1.