PaperlayerView exact revision on arXiv

Source-linked AI summary

Stochastic dynamics and the Polchinski equation: an introduction

Roland Bauerschmidt, Thierry Bodineau, Benoit Dagallier

arXiv:2307.07619v2math.PRmath-phmath.FAmath.MP

TL;DR

Functional inequalities help quantify the relaxation of stochastic dynamics, and this survey presents a renormalisation-group perspective on deriving them. It connects the Polchinski flow to stochastic localisation and optimal measure-generating processes, showing how the flow decomposes entropy by scale.

  • Problem

    How can functional inequalities quantify the relaxation of stochastic dynamics, including in high-dimensional statistical-mechanics models and their continuum limits?

    Source: §1 · p. 2

  • Method

    The survey uses the renormalisation-group flow of Gibbs measures and the Polchinski equation, relating its stochastic characteristics to Eldan’s localisation process and the Föllmer process.

    Source: §1 · pp. 2–3

  • Results

    The Polchinski flow coincides with an optimal stochastic process for a suitable varying metric, and its characteristics match those of Eldan’s stochastic localisation.

    Source: §1 · p. 3

  • Takeaways & Limitations

    In statistical-mechanics applications, the varying metric captures scale, giving the Polchinski flow a canonical way to decompose entropy by scale.

    Source: §1 · p. 3

    For the continuum sine-Gordon estimates discussed, control is obtained for β < 6π; extending it to the optimal regime β < 8π remains open.

    Source: Continuum sine-Gordon model · p. 47

Abstract

from arXiv · show

This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such as Eldan's stochastic localisation and the F\"ollmer process, the Bou\'e--Dupuis variational formula and the Barashkov--Gubinelli approach, the transportation of measure perspective, and the classical analogues of these ideas for Hamilton--Jacobi equations which arise in mean-field limits.

1 Introduction

The survey presents a renormalisation-group perspective on deriving log-Sobolev inequalities, motivated by the role of functional inequalities in understanding stochastic relaxation and by gaps for general continuous-spin systems. It connects the Polchinski flow to stochastic localisation and other stochastic and variational approaches.

1 Introduction

  • For general systems, especially continuous-spin systems, the picture of mixing properties and fast relaxation is less complete than for the Ising model.In the uniqueness regime, good mixing properties are broadly associated with fast relaxation through uniform Poincaré or log-Sobolev bounds.Source: §1 · p. 2
  • The survey uses renormalisation-group theory, in its continuous Polchinski formulation, to derive log-Sobolev inequalities from a multiscale generalisation of the Bakry–Émery criterion.The criterion is a condition on the renormalised potential, which obeys a second-order Hamilton–Jacobi-type equation.Source: §1 · p. 2
  • The Polchinski equation’s stochastic characteristics coincide with the process in Eldan’s stochastic localisation method, despite its different original purpose.The survey provides a dictionary relating the two viewpoints.Source: §1 · p. 3
  • Entropic stability offers an alternative to the multiscale Bakry–Émery method for the same Polchinski flow, while the discussed approaches are more spectral than spatially decomposed methods.The survey notes that spectral quantities can capture near-critical behaviour better, illustrated by applications.Source: §1 · p. 3
  • The Polchinski flow coincides with an optimal stochastic process for a suitable varying metric, which captures scale and decomposes entropy by scale.The renormalised potential can also be written as a Boué–Dupuis or Borell variational principle, a correspondence central to the Barashkov–Gubinelli method.Source: §1 · p. 3
  • The survey compares these results with control theory for classical Hamilton–Jacobi equations and briefly with the research line initiated in.These comparisons are presented in Appendix A.Source: §1 · p. 3

2 Background on stochastic dynamics

The section introduces continuous and discrete spin models with reversible Glauber–Langevin dynamics and develops entropy-based methods for quantifying relaxation toward equilibrium, including convexity criteria and entropy decomposition leading to log-Sobolev bounds. It then outlines challenges from interacting spin systems and continuum limits, where relaxation varies across scales and large scales contribute most to the log-Sobolev constant.

2.1. Motivation: Spin models and their stochastic dynamics.

  • The survey targets long-time dynamics of finite spin systems, including lattice limits and continuum limits related to singular SPDEs.Spin fields may take values in R or {±1}; in continuum approximations, a suitably normalized field describes the limiting SPDE solution.Source: §2.1 · p. 3, Continuous spins · p. 4

Continuous spins.

  • Continuous-spin Gibbs measures use a Hamiltonian with interactions and a local potential, and their reversible Glauber–Langevin dynamics are studied at large scales.A standard nearest-neighbor interaction and the Ginzburg–Landau–Wilson φ4 potential are examples.Source: Continuous spins · pp. 3–4

Discrete spins.

  • For discrete Ising spins, local single-spin flip rates satisfying detailed balance make the Gibbs measure invariant, and the focus is their large-time behavior as the system grows.The finite-state dynamics may be continuous- or discrete-time.Source: Discrete spins · p. 4

2.2. Generalities on Glauber–Langevin dynamics.

  • The dynamics evolve observables through a Markov semigroup, while reversibility yields an integration-by-parts identity and an associated Dirichlet form; ergodicity is assumed for convergence to equilibrium.In finite systems, the text describes this ergodicity as qualitative and linked to irreducibility.Source: Continuous spins · pp. 4–5

2.3. Log-Sobolev inequality.

  • The de Bruijn identity makes entropy nonincreasing along both continuous and discrete dynamics, while a log-Sobolev inequality yields exponential entropy decay and quantifies relaxation speed.In discrete state spaces, the modified log-Sobolev inequality gives exponential entropy decay, but the standard log-Sobolev inequality is stronger.Source: §2.3 · pp. 7–8
  • The log-Sobolev inequality is equivalent to semigroup hypercontractivity and implies a spectral-gap inequality with the same constant.The stated hypercontractivity equivalence does not hold in the same form for the modified log-Sobolev inequality used for discrete dynamics.Source: §2.3 · p. 8

2.4. Bakry–´Emery theorem.

  • If the continuous-spin Hamiltonian has Hessian bounded below by λ times the identity, the Bakry–Émery theorem gives a log-Sobolev constant γ ≥ λ.This criterion applies to strictly convex potentials; for quadratic Hamiltonians, the text states γ = λ.Source: §2.4 · pp. 8–9

2.5. Decomposition and properties of the entropy.

  • Conditioning splits entropy into a conditional entropy term for a simpler measure and a remaining term.The first term conditions on φ2 and concerns integration only over coordinate φ1.Source: §2.5 · p. 10
  • For Gibbs measures, the useful entropy decomposition depends on the measure’s structure; the renormalisation group provides a framework for this choice.The paper focuses on Gibbs measures arising in statistical mechanics and bases its method on the Polchinski equation, a continuous version of the renormalisation group.Source: §2.5 · p. 10
  • For product measures, log-Sobolev constants tensorise as the minimum of the factor constants, while general conditional entropy terms require additional bounds.The second conditional entropy term can be controlled by the desired Dirichlet form or bounded by κ Entν(F) with κ < 1, incurring an overall factor (1 − κ)^−1.Source: §2.5 · p. 10

2.6. Difficulties arising from statistical physics perspective.

  • In large nonconvex spin systems, phase transitions can change correlations, concentration, and Glauber-dynamics convergence, while standard log-Sobolev criteria become ineffective.Lack of convexity precludes the Bakry–Émery criterion, and the Holley–Stroock criterion is ineffective in large dimension when β > 0.Source: §2.6 · p. 12
  • Near criticality, a diverging correlation length is associated with a diverging inverse log-Sobolev constant, while product-case perturbation approaches can obscure its precise divergence.Renormalisation schemes for static features near βc instead use the Gaussian free field as a reference for entropy decomposition.Source: §2.6 · p. 12

2.7. Difficulties arising from continuum perspective.

  • For continuum φ4 limits, constructing the limit requires counterterm tuning in d ≥ 2, and the resulting microscopic nonconvexity rules out standard criteria.Although the limit can be constructed for d < 4, its regularisation parameter ε is not expected to affect the model’s physics; renormalisation arguments compare with the Gaussian free field.Source: §2.7 · p. 13
  • In the free-field dynamics, small Fourier scales equilibrate quickly while large scales relax slowly and dominate the log-Sobolev constant; the Polchinski flow replaces Fourier decomposition for interactions.Local real-space interactions do not align with the Fourier decomposition, motivating a smoother scale decomposition in the Polchinski flow.Source: §2.7 · p. 14

3 Gaussian integration and the Polchinski equation

The section develops Gaussian integration along a covariance decomposition as the basis for continuous, scale-by-scale renormalisation. Choosing that decomposition to reflect a model’s modes separates scales, which matters especially when integrating some scales changes the interactions among the rest.

3.1. Gaussian integration.

  • The Gaussian semigroup property follows from adding covariance matrices, matching the fact that independent Gaussian variables sum to a Gaussian with summed covariance.The measure decomposition is the continuous counterpart of splitting a Gaussian variable into independent components.Source: §3.1 · pp. 14–15
  • A covariance decomposition represents Gaussian integration continuously, and its parameter interval can be reparametrised without changing the constructions.The covariance path C_t increases continuously to C_∞ and is given by integrating its derivative.Source: §3.1 · p. 15
  • The Gaussian convolution F_t = P_{C_t} * F evolves by a heat equation with a time-dependent generator.The equation is interpreted weakly when C_t is not strictly positive definite.Source: §3.1 · p. 15
  • A useful covariance decomposition should reflect the model’s mode structure so scales can be integrated in order, especially when integrating some scales renormalises interactions among the rest.The decomposition choice depends on the model’s spatial structure and application.Source: §3.1 · pp. 14–15

3.2. Renormalised potential and Polchinski equation.

The Polchinski flow defines renormalised potentials, measures, and operators by integrating Gaussian fluctuations along a covariance decomposition. Its semigroup gives a successive-conditioning decomposition of the original measure, while the potential evolves by the Polchinski equation and the operators form a time-dependent Markov semigroup.

3.2. Renormalised potential and Polchinski equation.

  • The flow is considered for measures supported on a linear subspace, including affine subspaces by translation; the initial potential is assumed bounded below.The section points to a later section for generalisations to nonlinear spaces.Source: §3.2 · pp. 16, 18
  • The original measure’s expectations equal expectations under the renormalised measure after applying the Polchinski semigroup to the test function.This identity expresses the flow as a successive-conditioning decomposition.Source: §3.2 · pp. 16–17
  • The covariance split assigns local interactions to fast fluctuation scales and the long-range interaction to the slower variable, with the renormalised measure expected to have better properties.The conditional measure has a more convex Hamiltonian because its covariance is C_t ≤ C_∞.Source: §3.2 · p. 17
  • Because phase-transition models can involve many modes, renormalisation recursively integrates fast scales to simplify the measure step by step.Nested integrations give the semigroup property; controlling the evolving renormalised measure is also required.Source: §3.2 · p. 17
  • The renormalised potential V_t satisfies the Polchinski equation whenever C_t is differentiable.It is defined through the negative logarithm of the Gaussian convolution of e^-V_0.Source: §3.2 · pp. 17–18
  • The Polchinski semigroup operators form a time-dependent Markov semigroup, preserving positivity and constants and satisfying the composition law.The semigroup is generated by operators obtained by linearising the Polchinski equation.Source: §3.2 · p. 18

3.3. Log-Sobolev inequality via a multiscale Bakry–´Emery method.

The Polchinski flow yields a multiscale criterion for a log-Sobolev inequality based on renormalised potentials, rather than requiring the initial potential to be convex. Its proof adapts the Bakry–Émery strategy to the flow and decomposes entropy across scales.

3.3. Log-Sobolev inequality via a multiscale Bakry–´Emery method.

  • Theorem 3.6 gives a log-Sobolev inequality when an integrated Hessian bound on the renormalised potentials satisfies its multiscale condition; the initial potential need not be convex.When the initial potential is convex, the usual Bakry–Émery criterion is recovered; otherwise, the analysis of the renormalised potential is model dependent.Source: §3.3 · p. 19
  • The multiscale condition is invariant under smooth reparametrisations of time, and the estimates can also be formulated on a finite time interval.An alternative criterion avoiding a second derivative of the covariance is noted for decompositions where that derivative does not exist everywhere.Source: §3.3 · pp. 19–20
  • The proof adapts the Bakry–Émery argument to the Polchinski flow, using entropy production and a Hessian identity to establish the required estimate.The proof also imposes a technical continuity assumption on bounded smooth functions.Source: §3.3 · pp. 19–21
  • The flow decomposes entropy across successive scales; the discrete picture uses small increments, while the proof takes the limit as their size tends to zero.For small increments, the associated Gaussian covariance is tiny and the corresponding measure is extremely convex.Source: §3.3 · p. 21

3.4. Derivatives of the renormalised potential.

The section gives identities for derivatives of the renormalised potential and explains how its second derivative can be studied through the fluctuation measure. It also establishes preservation of convexity and recovers the standard Bakry–Émery criterion.

3.4. Derivatives of the renormalised potential.

  • The gradient and Hessian of the renormalised potential evolve by ∂tUt = LtUt and ∂tHt = LtHt − Ht ˙CtHt.These identities provide general tools for analysing derivatives, although their model-specific analysis may vary.Source: §3.4 · p. 22

The second derivative is encoded by a variance under the fluctuation measure

  • If V0 is convex, then Vt remains convex for every t ≥ 0 along the Polchinski flow.The section also gives a proof using log-concavity of a joint density and the Prékopa theorem.Source: The second derivative is encoded by a… · p. 23
  • Theorem 3.6 recovers the standard Bakry–Émery criterion when Hess H ≥ λid, Hess V0 ≥ 0, and the covariance decomposition is chosen as specified.The cited passage gives the resulting log-Sobolev constant, but its expression is truncated in the supplied text.Source: The second derivative is encoded by a… · p. 23
  • The second derivative of the renormalised potential is encoded by a variance under the fluctuation measure, which can be bounded using the Brascamp–Lieb inequality.The cited variance estimate is stated under the condition that Ct is invertible.Source: The second derivative is encoded by a… · p. 23

3.5. Example: Convexification along the Polchinski flow for one variable.

The one-variable example illustrates how the Polchinski flow smooths a non-convex initial potential and can improve convexity. A lower bound on the evolved curvature can then be used in the multiscale criterion to bound the initial measure’s log-Sobolev constant.

3.5. Example: Convexification along the Polchinski flow for one variable.

  • In this scalar case, the Polchinski equation is equivalent to a Hamilton–Jacobi–Bellman equation because e−Vt is a heat-kernel convolution of e−V0.The flow’s scale interpretation here concerns fluctuation size, rather than lattice geometry.Source: §3.5 · p. 25
  • For the non-convex one-variable example, convolution with the heat kernel blurs small-scale variation in V0, so Vt varies more slowly and retains larger-scale features.Here the only notion of scale is the size of field fluctuations.Source: §3.5 · p. 25
  • The evolved potential can become more convex: a uniform lower bound on its second derivative confirms that νt becomes more log-concave as t approaches 1.The example starts with a potential that is strictly convex outside a segment but has negative second derivative somewhere.Source: §3.5 · p. 26
  • A lower bound on the evolved curvature can be inserted into Theorem 3.6 to bound the log-Sobolev constant of ν0.The section notes that this illustrates the criterion’s effectiveness in non-convex cases, where standard Bakry–Émery and Holley–Stroock bounds may not yield dimension-independent estimates in large state spaces.Source: §3.5 · p. 26

3.6. Aside: Geometric perspective on the Polchinski flow.

The geometric perspective treats the covariance as an inverse metric and extends the flow to time-dependent, non-flat metrics. The resulting log-Sobolev criterion is conditional on a continuity assumption that may be difficult to verify.

3.6. Aside: Geometric perspective on the Polchinski flow.

  • The infinitesimal covariance ˙Ct plays the role of an inverse metric, and the construction extends the Polchinski equation to a time-dependent Riemannian metric.In the lattice heat-kernel example, the metric’s unit ball corresponds to heat-kernel smoothing of elements of the Euclidean unit ball.Source: §3.6 · pp. 26–27
  • The non-flat extension yields a log-Sobolev condition by adapting the proof of Theorem 3.6, provided the continuity assumption (3.99) holds.The construction defines a semigroup and a renormalised measure that evolves dually to it.Source: §3.6 · p. 28
  • In the non-flat case, the evolution of Zt is generally not explicit, making the required continuity assumption difficult to verify.Source: §3.6 · p. 28
  • The authors report no interesting applications of Theorem 3.20 beyond Theorem 3.6, while noting that finding such applications would be of interest.Source: §3.6 · p. 28

3.7. Aside: Entropic stability estimate.

The section presents entropic stability as an alternative route to log-Sobolev inequalities along the Polchinski flow, linking covariance control of exponential tilts to entropy contraction. Its bound applies for positive starting scales; the limit at zero requires a separate annealing step.

3.7. Aside: Entropic stability estimate.

  • Covariance control of all exponential tilts yields the entropic stability bound (3.105), which the section uses to derive entropy contraction and a possibly modified log-Sobolev inequality.The required covariance condition is stated in Lemma 3.21; the section identifies entropic stability as an alternative to the Bakry–Émery method for the same flow.Source: §3.7 · pp. 28–29
  • The contraction derived from entropic stability holds for s > 0; taking s → 0 in (3.109) is not valid, so annealing via a localization scheme supplies the final step.One stated route uses that the covariance C_s vanishes as s → 0, making the fluctuation measure uniformly log-concave, and then applies the Bakry–Émery criterion.Source: §3.7 · pp. 29–30
  • The entropic-stability strategy contracts expected entropy for the fluctuation measure, whereas the multiscale Bakry–Émery criterion contracts entropy for the renormalised measure.The two estimates express their right-hand sides through different terms in the entropy decomposition.Source: §3.7 · p. 30
  • For probability densities with a prescribed mean, exponential tilts minimize relative entropy; if no matching tilt exists, the relative entropy is infinite.The matching mean is E_µ[Fζ] = E_{T_hµ}[ζ].Source: §3.7 · p. 32

4 Pathwise Polchinski flow and stochastic localisation perspective

The section represents the Polchinski semigroup by a backward stochastic process, then uses that representation for coupling examples and martingale formulas. After reversing time and reparametrising, the process is identified with stochastic localisation, which emphasizes the fluctuation measure.

4.1. Pathwise realisation of the Polchinski semigroup.

  • For s ≤ t, the Polchinski semigroup is the conditional expectation of F(φ_s) given φ_t = φ, and the backward evolution carries ν_t to ν_s.The fluctuation measure at time t is the conditional distribution of the backward process at time 0 given φ_t.Source: §4.1 · p. 33

4.2. Example: Log-Sobolev inequality by coupling.

  • The log-Sobolev proof by coupling uses two backward processes driven by the same Brownian motion and applies the multiscale Bakry–Émery assumption to compare them.The section presents this as an alternative proof of Theorem 3.6 by synchronous coupling.Source: §4.2 · p. 35

4.3. Example: Coupling with the Gaussian reference measure.

  • At t = 0, the stochastic representation couples the full interacting field φ_0 with the Gaussian reference field Γ_0.The coupling is obtained from the stochastic realisation of P_{t,∞}.Source: §4.3 · p. 35

4.4. Renormalised potential and martingales.

  • The renormalised potential, its gradient, and its Hessian have stochastic representations; the gradient along the backward process is a martingale.The martingale statements are with respect to the backward filtration; the Hessian evolution includes the term −H_s ˙C_s H_s.Source: §4.4 · p. 36

4.5. Stochastic localisation perspective.

  • After reversing time and reparametrising, the stochastic evolution coincides with Eldan’s stochastic localisation process, whose focus is the fluctuation measure.In that perspective the twisted measure becomes uniformly convex with Hessian at least t when the starting potential is log-concave, enabling concentration inequalities to transfer to the initial measure; increasing t trades improved convexity against loss of original-measure features.Source: §4.5 · pp. 37–38

5 Variational and transport perspectives on the Polchinski flow

The survey connects the Polchinski flow to optimal stochastic control and transport, using these perspectives to relate renormalised measures to entropy and functional inequalities. It presents optimal-drift characterisations and Lipschitz transport results under specified covariance decompositions and assumptions.

5.1. F¨ollmer’s problem.

  • The gradient of the renormalised Polchinski potential is the optimal drift, and its path-space entropy equals the endpoint relative entropy H(ν0|γ0).The path-space relative entropy is H(Q|P), where Q is the process law and P the Gaussian reference-process law.Source: §5.1 · pp. 39–40
  • For targets Fν0, the optimal drift is expressed through the Polchinski semigroup, recovering the entropy decomposition.The section treats the Dirac-start case of the broader Schrödinger problem and notes that other initial measures are possible in that general problem.Source: §5.1 · p. 40

5.2. Variational representation of the renormalised potential.

  • The Boué–Dupuis or Borell formula gives equality in the variational representation of the renormalised potential and underlies the Barashkov–Gubinelli method.Estimating the representation’s expectation for a general drift can yield bounds on Vt and V∞.Source: §5.2 · p. 41, Sketch · p. 41

5.3. Lipschitz transport.

  • Transport estimates can transfer log-Sobolev inequalities from the rescaled measures to ν0, and the same argument applies to more general functional inequalities.The alternate covariance-decomposition result controls the map’s gradient in a t-dependent input norm, with a Gaussian log-Sobolev inequality used to compensate for the loss of regularity.Source: §5.3 · pp. 41–43
  • Rescaling the degenerating Polchinski measures produces measures comparable to ν0, and the rescaled measures converge in distribution to a Gaussian measure.The stated limit is PC∞ = PA^-1.Source: §5.3 · pp. 42–43
  • Under the multiscale Bakry–Émery criterion, the covariance choice Ċt = e^-tA yields an exp(-1/2 μt)-Lipschitz transport map; finite integrated μt gives a corresponding map from the Gaussian measure to ν0.The section also extends the argument to Ċt = (tA + id)^-2.Source: §5.3 · p. 43

6 Applications

The section applies renormalisation-based criteria and potential bounds to statistical-mechanics models and dynamics, including Ising models, transport maps, alternative Dirichlet forms, and conservative dynamics. It develops log-Sobolev and related bounds, examining uniformity in volume and regularisation, behaviour near criticality and uniqueness thresholds, and conditions for transferring the method to other dynamics.

6 Applications

  • At high temperature the criterion generally gives volume-uniform log-Sobolev bounds, and examples show it can remain effective up to phase transitions.For statistical-mechanics models, the Ising case is an example of the high-temperature result; the criterion reaches criticality in the Ising and φ4 examples.Source: §6 · p. 46

Continuum sine-Gordon model.

  • For massive sine-Gordon in d = 2, β < 6π yields a log-Sobolev lower bound uniform in ε; under a small-coupling condition, it is also uniform in L.The large-volume condition is L ≥ 1/m and |z| ≤ δβm^(2+β/4π); the stated bound is greater than m² − Oβ(…).Source: Continuum sine-Gordon model · p. 48
  • For g > 0 and r > rc(g), the log-Sobolev constants obey inf over ε,L γε,L(g,r) > 0; at fixed L and any r ∈ R, inf over ε γε,L(g,r) > 0.The dependence on g and r is accurate only far from transition (r > 0 and g ≪ r); near criticality, polynomial bounds are stated for lattice φ4 in d > 4 and the Ising model.Source: Continuum sine-Gordon model · pp. 49–50

6.2. Applications to lattice φ4 models.

  • For lattice φ4 in any dimension, the criterion bounds the log-Sobolev constant for r > rc(g); under a mean-field susceptibility bound, it gives polynomial control in 1/δ.The near-critical susceptibility assumption applies in particular to φ4 on Zd for d ≥ 5, with δ = L^−d + r − rc(g) for r ≥ rc(g).Source: §6.2 · p. 50

Sketch of Examples 6.3–6.4.

  • The lattice φ4 bounds follow by controlling the susceptibility integral: a uniformly bounded susceptibility yields a positive log-Sobolev lower bound, while the mean-field bound gives polynomial control.The proof combines a Brascamp–Lieb estimate at sufficiently large shifted r with monotonicity of susceptibility from the second Griffiths inequality.Source: Sketch of Examples 6.3–6.4 · pp. 50–51

6.3. Applications to transport maps.

  • For the continuum sine-Gordon model, transport-map Lipschitz constants are uniform in t and L; for continuum φ4, the map gradient is uniformly bounded from H1 to L2.The sine-Gordon bound follows under Theorem 6.1’s assumptions; the φ4 bound uses Theorem 6.2’s assumptions and yields the log-Sobolev inequality for the continuum φ4 measure.Source: §6.3 · p. 51, Sketch · p. 51

Sketch.

  • For lattice φ4 with r > rc(g), the transport map is uniformly Lipschitz in L and t; under the mean-field bound, its Lipschitz constant is of order δ^-D/2.The stated bound is C(g, D)δ^-D/2.Source: Sketch · p. 52

6.4.1. Renormalised potential.

  • For Ising models, the renormalised field is continuous as soon as t > 0, so convexity criteria can yield log-Sobolev inequalities despite the discrete spins.The Polchinski equation remains applicable and the renormalised potential becomes smooth immediately.Source: §6.4.1 · pp. 52–53

Ising model, standard Dirichlet form.

  • For a single spin, the standard Dirichlet-form log-Sobolev constant is at least 2, while the heat-bath modified log-Sobolev constant is at least 1/2.These single-spin inequalities provide product-measure input for the Ising arguments.Source: Ising model, standard Dirichlet form · p. 53

6.4.3. Entropy decomposition.

  • The entropy decomposition splits the Ising measure into an infinite-temperature product measure and a renormalised measure, with the two entropy terms controlled using their respective log-Sobolev inequalities.The same strategy can accommodate other jump rates and modified log-Sobolev or spectral inequalities.Source: §6.4.3 · p. 54

Sketch.

  • For β<1, the Ising model satisfies a log-Sobolev inequality under the spectral high-temperature condition.The coupling matrix is assumed to have spectrum in [0, 1].Source: Sketch · p. 55

6.4.4. Hessian of the renormalised potential and covariance.

  • For β>1, multiscale Bakry–Émery and entropic-stability approaches yield covariance-condition bounds; for ferromagnetic models these apply up to criticality and give polynomial log-Sobolev bounds under the mean-field condition in d ≥ 5 [31].The renormalised-potential Hessian is controlled by covariance estimates, and for ferromagnetic interactions the covariance spectral radius is maximal at zero field.Source: Sketch · pp. 55–56, §6.4.4 · pp. 56–57
  • For ferromagnetic Ising models, the log-Sobolev constant is uniformly bounded below in lattice size below βc; susceptibility bounds yield polynomial estimates near and at criticality, including in dimensions d ≥ 5.For mean-field interactions, the bound has the correct order βc − β. The polynomial degree is not expected to be sharp unless D = 1; the nearest-neighbour susceptibility assumptions hold for d ≥ 5.Source: §6.4.4 · pp. 57–58
  • On graphs of maximal degree d, the ferromagnetic Ising log-Sobolev constant is uniformly bounded below when (d−1) tanh β < 1, with polynomial divergence toward βu and in graph size at βu.Here βu = artanh(1/(d − 1)); this improves the condition β < 1/(2d) from (6.66), and is the uniqueness threshold on the infinite d-regular tree.Source: §6.4.4 · p. 58

Sketch.

  • Comparing susceptibility with a d-regular tree gives a polynomial lower bound for the graph-size dependence of the log-Sobolev constant at βu.The susceptibility estimate combines its tree comparison with the finite-graph bound χβ ≤ N [93].Source: Sketch · p. 58

6.4.5. Choice of Dirichlet form.

  • Comparing reversible jump rates within a factor K transfers log-Sobolev constants, but unbounded random couplings can make this comparison ineffective, motivating direct bounds for other dynamics.The arguments also extend beyond the standard Dirichlet form when the single-spin inequality is uniform in the field and the form is concave in the measure.Source: §6.4.5 · pp. 58–59
  • For the heat-bath Dirichlet form, the modified log-Sobolev constant satisfies the same bound as for the standard form, up to a factor of 4 from normalization.The argument uses the uniform single-spin modified inequality and concavity of Ψ(a, b) = ab/(a+b).Source: §6.4.5 · p. 59

6.5. Applications to conservative dynamics.

  • Theorem 3.6 applies in principle to conservative dynamics, including spin models with constrained magnetisation; related results cover discrete and continuous spins and the continuum sine-Gordon model [18].The continuous-spin example uses a potential strictly convex outside a segment and Brownian motion on the zero-magnetisation hyperplane.Source: §6.5 · pp. 59–60

A.1. Hamilton–Jacobi equation.

This section develops classical Hamilton–Jacobi analogues of the paper’s stochastic constructions, using action minimisers to parallel renormalised measures. It also contrasts the equations’ regularity and scope.

A.1. Hamilton–Jacobi equation.

  • The minimum action is V1(0), paralleling the renormalised potential; the constructions have Hamilton–Jacobi analogues.Source: §A.1 · p. 60
  • Hamilton–Jacobi solutions can develop shocks and need only be locally Lipschitz, unlike the smooth finite-variable Polchinski solutions.In the thermodynamic limit, shocks can also form in the Polchinski equation and correspond to phase transitions.Source: §A.1 · p. 60
  • The classical constructions are presented to emphasise their analogy with the stochastic constructions and to offer another intuition, under convenient regularity assumptions.Source: §A.1 · p. 60
  • Hamilton–Jacobi characteristics follow the velocity field ∇Vt, but the corresponding φ need not be unique when S has multiple minimisers or V is not globally smooth.Source: §A.1 · p. 61

Sketch of Proposition A.1.

The sketch links action minimisers to Hamilton–Jacobi characteristics and gives a classical control analogue of Föllmer’s problem, with the renormalised-potential gradient as optimal drift.

Sketch of Proposition A.1.

  • An isolated local minimum of St forces the characteristic path to satisfy ˙φt = ∇Vt(φt).The proof combines the Euler–Lagrange equation with its time derivative; positive definiteness of the Hessian at the minimum yields the identity.Source: Sketch of Proposition A.1 · p. 61
  • In the classical Föllmer control problem, the optimal drift is ∇V, and the associated bound is attained for that choice.Source: Sketch of Proposition A.1 · p. 62
  • The classical path-space entropy analogue is the integrated squared-speed cost, 1/2 ∫ |˙φuU|² du.Source: Sketch of Proposition A.1 · p. 62

Sketch.

The sketch uses the Hopf–Lax formula to connect the Hamilton–Jacobi equation with a classical control bound; under its stated conditions, the renormalised-potential gradient is the optimal drift.

Sketch.

  • The Hopf–Lax formula supplies the bound, with equality when the control follows the gradient drift.Source: Sketch · pp. 62–63
  • The gradient of the renormalised potential Vt provides the optimal drift.Source: Sketch · p. 62
  • The result is stated under regularity conditions on V0 and for t ≤ 1; shocks can also prevent uniqueness of the minimiser.Source: Sketch · p. 63
  • The corresponding solution need not be unique.Source: Sketch · p. 63

A.2. Example: Mean-field Ising model.

The mean-field Ising example connects the finite-system Polchinski flow to Hamilton–Jacobi equations for the limiting renormalised potential and free energy. Constant-field reduction makes the limiting equation one-dimensional.

A.2. Example: Mean-field Ising model.

  • The Ising measure is rewritten using P = id − Q, where Q projects onto constant vectors, separating the constant-field direction.Source: §A.2 · p. 63
  • For constant external field h1, the limiting free energy is the viscosity solution of a Hamilton–Jacobi equation.Source: §A.2 · pp. 63–64
  • The renormalised potential satisfies the Polchinski equation, whose right-hand side depends only on derivatives in constant directions.Source: §A.2 · p. 64
  • As N → ∞, the reduced one-variable Polchinski equation converges to a unique viscosity solution of a one-variable Hamilton–Jacobi equation.The reduced equation has a prefactor 1/N on its Laplacian term.Source: §A.2 · p. 64
  • The Hamilton–Jacobi equation for the free energy follows from the one for the renormalised potential through their stated relation.Source: §A.2 · pp. 64–65

Related papers

Loading 2307.07619v2…