PaperlayerView exact revision on arXiv

Source-linked AI summary

High-Dimensional Ultra-Log-Concave Distributions

Zongchen Chen, Sihan Wang

arXiv:2609.23994v1math.PRcs.DS

TL;DR

The paper asks how to extend ultra-log-concavity to high-dimensional measures on the nonnegative lattice, where canonical local birth-death dynamics are difficult to analyze. It introduces δ-ULC and combines integrated Bakry–Émery methods with Poisson stochastic localization to derive functional inequalities and structural results. The theory includes characterizations of ULC, applications to several models, and a lattice scaling limit to Laguerre diffusions.

  • Problem

    For measures on unbounded subsets of N^d, existing finite-support homogenization does not directly analyze the canonical coordinatewise birth-death chain or expose its local curvature.

    Source: §1 · p. 4

  • Method

    The paper defines δ-ULC through local curvature and combines integrated discrete Bakry–Émery calculus for birth-death chains with Poisson stochastic localization.

    Source: §1 · p. 5, §1.4 · p. 11

  • Results

    δ-ULC yields a weighted Poincaré inequality, while ULC is characterized equivalently by strongly log-concave and completely log-concave probability-generating functions.

    Source: §1.1 · p. 6, §1.2 · p. 7

  • Takeaways & Limitations

    The framework supplies concentration, maximum-entropy and closure results for ULC measures, with applications including queueing, polymatroids, antiferromagnetic Potts models, and hardcore models.

    Source: §1 · p. 5, §5 · p. 46

    The equivalence involving Item 5 is not proved in the main argument; its proof is deferred to Section A.3 and requires homogenization and Lorentzian-polynomial theory.

    Source: §4.7 · p. 44

Abstract

from arXiv · show

Ultra-log-concave distributions are ubiquitous in probability, combinatorics, and statistical mechanics and have been studied extensively. In this paper, we introduce a quantitative high-dimensional extension of this notion, called $\delta$-ultra-log-concavity, for probability measures on $\mathbb{N}^d$ with downward closed support. When $\delta = 1$, this notion coincides with the class studied by Gurvits (2009) via strongly log-concave generating functions, and with the class defined by Anari, Oveis Gharan, and Vinzant (2021) via completely log-concave generating functions; in one dimension, it reduces to classical ultra-log-concavity. We establish several functional inequalities, including a weighted Poincar\'e inequality, a discrete Brascamp--Lieb inequality, and a weighted Wu-type modified log-Sobolev inequality. Our approach combines integrated Bakry--\'Emery calculus for a canonical birth-death chain with Poisson stochastic localization, which arises as the time reversal of coordinatewise binomial thinning. We further establish concentration of measure, maximum-entropy principles, and several closure properties for ultra-log-concave measures, and develop applications to queueing models, polymatroids, antiferromagnetic Potts models, and hardcore models. Finally, a lattice scaling limit of the discrete theory yields Poincar\'e and Brascamp--Lieb inequalities for Laguerre diffusions.

1 Introduction

The introduction presents an extension of ultra-log-concavity to probability measures on downward closed subsets of N^d, using integrated Bakry–Émery calculus and Poisson stochastic localization to derive functional inequalities and applications for canonical birth-death dynamics. It also situates these contributions in prior work and describes connections to generating functions and Laguerre diffusion.

1 Introduction

  • Local Markov chains are practical, but their global convergence is difficult to analyze; finite-support homogenization does not directly analyze the canonical birth-death process.The introduction motivates a local curvature condition for the original process, rather than a black-box reduction through truncation.Source: §1 · p. 4

1.1 High-Dimensional Ultra-Log-Concave Distributions

  • The ULC condition matches classical ultra-log-concavity in one dimension, the binary local-matrix condition, and is preserved by convolution, marginalization, binomial thinning, and stochastic projection.The same class is characterized by strong and complete log-concavity of the probability-generating function and by 1-spectral independence under tilted weighted pinnings.Source: §1.1 · p. 6

1.2 Functional Inequalities

  • Every δ-ULC measure has a dimension-free weighted Poincaré inequality and canonical birth-death spectral gap at least δ; the bound is sharp for exact δ-ULC measures.The weighted inequality also gives covariance control and (1/δ)-spectral independence with respect to Poisson stochastic localization.Source: §1.2 · p. 7
  • Positive local curvature yields a discrete Brascamp–Lieb inequality, while ULC measures with finite first moments satisfy a weighted Wu-type modified log-Sobolev inequality with sharp constant 1 for product Poisson measures.The Wu-type inequality requires ULC itself: the additional diagonal term for δ < 1 is not preserved under arbitrary exponential tilts.Source: §1.2 · pp. 7–8
  • The Wu-type inequality implies Poisson-type concentration and a maximum-entropy principle: among ULC measures with fixed mean, the product Poisson measure is the unique entropy maximizer.The results apply when the measure has finite positive mean.Source: §1.2 · pp. 8–9

1.3 Examples and Applications

  • Two interacting infinite-server queues are δ-ULC, with δ = 1 − |b_long − b_short| when |b_long − b_short| < 1; discrete polymatroid-conditioned product-Poisson measures are ULC and satisfy modified log-Sobolev inequalities with constant 1.When the queue arrival rates are equal, the queues are independent Poisson systems.Source: §1.3 · pp. 9–10
  • For antiferromagnetic Potts models, the downward-closed encoding gives δ-ULC under the stated condition and a heat-bath Glauber spectral-gap bound; for regular-graph hardcore models, Brascamp–Lieb yields an explicit variance bound.For random regular graphs and fixed q, the Potts result gives a polynomial mixing regime that includes, for large degree, parameters beyond the tree-uniqueness threshold.Source: §1.3 · pp. 10–11

1.4 Our Approach

  • The resulting weighted Wu-type inequality gives Poisson domination, concentration, and a maximum-entropy principle; lattice scaling also yields Laguerre Poincaré and Brascamp–Lieb inequalities.Under scaling, the generators, Dirichlet forms, and local curvature matrices converge to their continuous Gamma-reference counterparts.Source: §1.4 · pp. 11–12
  • An exact integrated discrete Bochner identity yields the weighted Poincaré inequality, while retaining its full matrix structure gives the discrete Brascamp–Lieb inequality.The argument uses Kn ⪰δDn and does not require a pointwise CD (δ, ∞) condition.Source: §1.4 · p. 11
  • Binomial-thinning localization on Nd reverses to the Poisson–Föllmer process, whose posterior measures form a linear-tilt flow with exact conditional Φ-entropy dissipation identities.ULC is preserved along the flow; covariance bounds yield approximate entropy conservation and the weighted Wu-type inequality.Source: §1.4 · pp. 11–12

1.5 Related Work

  • For measures on Nd, the paper proves ULC is equivalent to strong and complete log-concavity of the probability-generating function, extending prior characterizations.The characterization is stated in Theorem 4.26.Source: §1.5 · p. 12
  • The paper extends the integrated Bakry–Émery local matrix criterion from binary to count-valued configurations and gives that matrix a curvature interpretation [GJM+26].A point-process lift also recovers the earlier criterion for point-process Glauber dynamics.Source: §1.5 · pp. 12–13

1.6 Organization

  • The paper’s roadmap assigns functional inequalities and hardcore-model applications to Section 3, localization and log-Sobolev results to Section 4, and discrete-to-continuous scaling to Section 7.Sections 5 and 6 cover Poisson comparison and concentration, then polymatroids, Potts models, and algorithmic consequences.Source: §1.6 · p. 13

2 Preliminaries

The preliminaries establish notation for measures, vectors, and discrete differences, then define Markov-chain dynamics and the functional inequalities used to relate convergence to mixing.

2.1 Notation

  • Notation fixes conventions for vectors, measures, differences, supports, tilts, and common probability distributions.A downward closed set contains every nonnegative coordinatewise smaller point of each of its elements.Source: §2.1 · pp. 13–14

2.2 Discrete-Time Markov Chains and Continuous-Time Markov Processes

  • Markov chains and continuous-time processes are described by transition kernels or jump rates, with reversibility expressed through detailed balance.The section also introduces time-inhomogeneous transitions and the associated generator and Dirichlet form.Source: §2.2 · pp. 15–17

2.3 Functional Inequalities and Mixing Times

  • Poincaré and modified log-Sobolev inequalities yield variance and entropy contraction, respectively, and provide mixing-time bounds for finite reversible chains.The Poincaré constant is the spectral gap; the stated discrete-time bounds assume a positive-semidefinite kernel.Source: §2.3 · pp. 17–18

3 Bakry–´Emery Theory for Birth-Death Chains

This section develops integrated Bakry–Émery and curvature methods for reversible birth-death chains with linear death rates, deriving a discrete Bochner formula and weighted Poincaré and discrete Brascamp–Lieb inequalities. It also applies these inequalities to variance bounds for the hardcore model and gives a spectral-gap criterion for a two-queue model.

3.1 Bakry–Émery Calculus

  • For a finite irreducible reversible chain, the integrated Bakry–Émery bound with constant δ is equivalent to the Poincaré inequality with constant δ, and pointwise CD(δ, ∞) implies it.The integrated criterion also applies to pure-jump generators that lack the diffusion chain rule.Source: §3.1 · p. 19

3.2 Birth-Death Generators, Dirichlet Forms, and Poincaré Inequalities

  • The chain uses linear death rates d_i(n) = n_i and birth rates chosen by detailed balance with µ; its reversible Dirichlet form yields a weighted Poincaré inequality whose best constant is the spectral gap.The measure is assumed to have downward closed support.Source: §3.2 · p. 20

3.3 Integrated Bakry–Émery Criterion and Integrated Γ2 Form

  • The discrete Bochner formula expresses integrated Γ2 through a weighted discrete Hessian energy and a first-order form comparing neighboring ratios of a(n) = n!µ(n).The first term is nonnegative when the two functions coincide, and the remaining coefficients act as a discrete analogue of the curvature matrix.Source: §3.3 · p. 21

3.4 Ultra-Log-Concave Measures and Weighted Poincaré Inequalities

  • The δ-ULC condition requires a pointwise matrix inequality on active coordinates, defined using a(n) = n!µ(n), for measures with downward closed support.The condition is motivated by discarding the nonnegative discrete Hessian term in the Bochner formula.Source: §3.4 · pp. 23–24
  • Every δ-ULC probability measure on N^d satisfies the weighted Poincaré inequality with constant δ, equivalently giving its birth-death generator spectral gap at least δ.The result applies to measures with downward closed support.Source: §3.4 · p. 24
  • When the one-dimensional δ-ULC condition holds with equality at every supported step, µ is negative binomial for 0 < δ < 1, Poisson for δ = 1, or binomial for δ > 1.In these cases δ is the optimal weighted Poincaré constant, and the centered identity function is an eigenfunction with eigenvalue δ.Source: §3.4 · pp. 24–25
  • In one dimension, ULC is the classical condition that a(n) = n!µ(n) is log-concave.The paper notes that the resulting weighted Poincaré inequality recovers the classical discrete Poincaré inequality.Source: §3.4 · p. 25
  • A product measure on N^d is δ-ULC exactly when each factor is δ-ULC, and its birth-death generator’s spectral gap is the minimum of the factor spectral gaps.Source: §3.4 · p. 25

3.4 Ultra-Log-Concave Measures and Weighted Poincar´e Inequalities

  • For two queues with longer- and shorter-queue arrival rates blong and bshort, a positive ULC parameter exists exactly when |blong − bshort| < 1, yielding spectral gap at least 1 − |blong − bshort|.The chain has M/M/∞-type service, with departure rate ni from queue i; tied queues each receive arrivals at rate blong.Source: §3.4 · pp. 25–26

3.5 Local Curvature Matrix and Discrete Brascamp–Lieb Inequalities

  • For measures on Nd with downward closed support, the discrete Brascamp–Lieb inequality retains the full state-dependent local curvature matrix rather than only a uniform curvature lower bound.The resulting inequality recovers the weighted Poincaré inequality.Source: §3.5 · p. 27
  • For measures on Zd with upward closed support, the unit-birth-rate chain likewise yields a discrete Brascamp–Lieb inequality using the full local curvature matrix.Under the c-log-concavity condition, this result recovers the Poincaré inequality proved in.Source: §3.5 · pp. 28–29

3.6 Variance Bounds in the Hardcore Model

  • For a Δ-regular graph, the hardcore model’s variance fraction satisfies VG(λ) ≤ λ/[1 + (Δ + 1)λ], with the bound remaining finite at 1 + λ^-1 = λ⋆.Here λ⋆ = −λmin(AG), where AG is the graph’s adjacency matrix.Source: §3.6 · p. 30

4 Poisson Stochastic Localization

This section develops Poisson stochastic localization using coordinatewise binomial thinning and a limiting-process perspective to study ultra-log-concave measures. It connects the process to spectral independence and conservation of variance and entropy, and uses it to derive modified log-Sobolev inequalities.

4.1 The Localization-Scheme Framework

  • The localization framework uses posterior laws as conditional measures in a martingale, with approximate conservation of conditional variance and entropy furnishing functional-inequality bounds.The construction starts from a tractable noising process independent of the target measure; spectral independence and entropic stability can help establish the required conservation estimates.Source: §4.1 · pp. 31–32

4.2 Noising and Denoising Processes

  • Coordinatewise binomial thinning defines a death-process noising scheme whose time reversal is a birth process, with birth rates given by a conditional-expectation formula.The rate is qt(Xt) = 1/(1 − t) E[X1 − Xt | Xt].Source: §4.2 · pp. 33–34

4.3 Localization Process

  • Poisson stochastic localization is a linear-tilt scheme driven by a compensated-birth martingale, and its conditional Φ-entropies satisfy exact dissipation identities.Variance and entropy are important special cases of these conditional Φ-entropies.Source: §4.3 · pp. 35–36

4.4 Associated Limiting Process

  • As t approaches 1, accelerating the reversible noising-denoising kernels by 1/(1 − t) yields the birth-death chain, and the corresponding functional inequalities pass to the limit.At finite t, the kernel has Poincaré constant γt, which is optimal, and modified log-Sobolev constant ρt.Source: §4.4 · pp. 37–38
  • For downward-closed support and finite first moments, the limiting birth-death chain yields weighted Poincaré and Wu-type modified log-Sobolev inequalities for the target measure.The inequalities follow by taking the accelerated noising-denoising kernels to their limit as t approaches 1.Source: §4.4 · pp. 37–38
  • For the limiting process L, the functional inequalities extend to their stated domains by showing compactly supported functions form cores for the relevant Dirichlet form and operator.The extension uses truncation and restriction to coordinate boxes, with boundary contributions vanishing under the finite-first-moment assumption.Source: §4.4 · p. 39

4.5 Conservation of Variance and Entropy via Spectral Independence

  • For downward-closed support, η-spectral independence is equivalent to the weighted Poincaré inequality on linear functions with constant 1/η, and every δ-ULC measure is 1/δ-spectrally independent..Source: §4.5 · pp. 40–41
  • Spectral independence of the tilted pinnings yields differential inequalities expressing approximate conservation of conditional variance and entropy along Poisson stochastic localization.Proposition 4.20 gives the variance bound when (1 − t) ∗ µ_x is η_t-spectrally independent for every pinning x; its entropy bound applies when all positive tilts of the pinnings are η-spectrally independent.Source: §4.5 · p. 41

4.6 Modified Log-Sobolev Inequalities for ULC Measures

  • Every ULC measure with finite first moments satisfies the weighted Wu-type modified log-Sobolev inequality for the limiting process with constant 1.The proof uses preservation of ULC under weighted pinning and exponential tilting, followed by entropy conservation and transfer to the limiting process.Source: §4.6 · pp. 41–42
  • In one dimension, the weighted inequality recovers the Wu-type modified log-Sobolev inequality, whose sharp constant is |log µ(0)|, refuting the conjectured improvement to mean(µ).The paper also notes that this recovers [Joh17, Theorem 1.3].Source: §4.6 · p. 43

4.7 ULC Measures and Log-Concave Probability-Generating Functions

  • For measures on N^d with downward-closed support, ULC is characterized by 1-spectral independence of tilted pinnings and strong or complete log-concavity of the probability-generating function [Gur09a] [AOV21].The theorem states these conditions are equivalent; the paper proves the equivalence of items 1–4 and defers the 1 ⇒ 5 implication for complete log-concavity. This extends [CCC+25, Proposition 3.8] beyond the hypercube.Source: §4.7 · p. 44
  • The limiting process L satisfies a modified log-Sobolev inequality with constant 1 for any finite-first-moment measure with a strongly log-concave probability-generating function.This conclusion does not require downward-closed support.Source: §4.7 · p. 45
  • The support condition changes the form of L: it is the birth-death process for downward-closed support and a continuous-time down-up walk for homogeneous support.The passage identifies the homogeneous-support case but does not provide further details about its precise support condition.Source: §4.7 · p. 45
  • The result extends to N^d and supplies a new intrinsic proof based on Poisson stochastic localization.Source: §4.7 · p. 45
  • Strong log-concavity of the probability-generating function forces the measure's support to be M♮-convex.Source: §4.7 · p. 45

5 Poisson-Type Concentration and Maximum Entropy

The section derives Poisson-type concentration and maximum-entropy consequences from a weighted modified log-Sobolev inequality, including extensions beyond ULC measures. Its key comparison bounds moment-generating functions by those of a product Poisson law.

5 Poisson-Type Concentration and Maximum Entropy

  • The section also extends the monotone entropic law of thin numbers and entropy concavity under thinning to high-dimensional ULC measures.The proofs are omitted because they require only minor modifications of one-dimensional arguments.Source: §5 · p. 46

5.1 Poisson Moment-Generating Function Domination

  • Theorem 5.1 bounds the moment-generating function of every WLSI (1) measure by that of the product Poisson law with the same mean.The comparison extends the one-dimensional ULC result of [AMM22, Lemma 2.1] to WLSI (1).Source: §5.1 · p. 46

5.2 Poisson-Type Concentration Inequalities

  • The bound yields mixed upper- and lower-tail Poisson-type concentration estimates for WLSI (1) measures.For one-dimensional ULC measures, the resulting bounds recover [AMM22, Theorem 1.1].Source: §5.2 · p. 48

5.3 Poisson Maximum-Entropy Principle

  • Among WLSI (1) measures with mean m, the product Poisson law is the unique maximizer of Shannon entropy.The entropy inequality includes a relative-entropy remainder; ULC measures with finite first moments belong to WLSI (1).Source: §5 · p. 46, §5.3 · p. 49

6 Examples

The examples apply the local-curvature criterion to polymatroid measures and embedded antiferromagnetic Potts measures. For polymatroids, the resulting ULC property also gives functional-inequality and sampling consequences.

6 Examples

  • The examples section separates model-specific local matrix calculations from the general functional-inequality machinery.The stated scope includes establishing ULC for polymatroid and Potts models and deriving algorithmic consequences.Source: §6 · p. 50

6.1 Weighted Independent Vectors in Polymatroids

  • For every integral polymatroid rank function and positive activity vector, the Poisson-weighted measure on its independent vectors is ULC.Its local curvature matrix decomposes into all-ones and zero blocks, which are positive semidefinite.Source: §6.1 · pp. 50–51
  • The polymatroid measure's ULC property implies complete log-concavity of its probability-generating function and supports a sampler with a modified log-Sobolev bound.The lazified chain has constant at least 1/(2C); its mixing bound is stated in the supplied theorem passage.Source: §6.1 · pp. 51–52

6.2 Antiferromagnetic q-Potts Model

  • For graphs of maximum degree Δ under the stated condition, the downward-closed embedding of the antiferromagnetic q-state Potts measure is δ-ULC.The theorem assumes q ≥ 3 and 0 < B < 1, with δ given by its stated curvature bound.Source: §6.2 · p. 52
  • For sufficiently large Δ, the embedded Potts measure on a uniformly random Δ-regular graph is δ-ULC with δ ≥ 1/2 with high probability.For fixed q, this regime includes B below the conjectured tree-uniqueness threshold for sufficiently large Δ.Source: §6.2 · p. 54

7 Laguerre Dynamics

The discrete linear-death chain converges under lattice scaling to Laguerre diffusion, and the discrete curvature condition scales to a Laguerre curvature condition. This yields weighted Poincaré and Brascamp–Lieb inequalities in the continuous setting.

7 Laguerre Dynamics

  • The scaling limit of the linear-death chain is Laguerre diffusion, with the generator converging to the Laguerre generator.The pushforward measures converge by Riemann-sum convergence.Source: §7 · pp. 54–55
  • The discrete δ-ULC condition scales to the κ-Laguerre-ULC condition under the stated curvature limit.With uniform Taylor-remainder control, for every κ′ < κ one obtains δN with NδN → κ′ and µN δN-ULC.Source: §7 · pp. 55–56
  • For κ-Laguerre-ULC measures, Theorem 7.1 gives weighted Poincaré and Brascamp–Lieb inequalities.The theorem also states a Brascamp–Lieb inequality when the local curvature matrix is positive definite everywhere.Source: §7 · p. 56
  • Product Gamma measures are exactly κ-Laguerre-ULC and attain the sharp weighted Poincaré constant κ.The coordinate functions specified in the passage are eigenfunctions with eigenvalue κ.Source: §7 · p. 57

AI Disclosure

  • The authors disclose AI assistance in literature searching, proof development, exposition, and typesetting, and state that they independently verified all content.The disclosure distinguishes proofs developed by the authors from those assisted by named AI tools.Source: AI Disclosure · p. 58

A.1 Itˆo Calculus for Poisson Stochastic Localization

The appendix derives entropy identities for Poisson stochastic localization using martingale and jump-process calculus. The identities specialize to variance and entropy, while integrability is controlled on intervals ending before time 1.

A.1 Itˆo Calculus for Poisson Stochastic Localization

  • The stochastic integral defining Zt,i is integrable and is a martingale on [0, T].For T < 1, the relevant compensator integral is bounded by -log(1-T).Source: §A.1 · p. 64
  • Applying the jump-process Itô formula to the martingales Ft and Gt gives a general Φ-entropy identity.The derivation uses coordinate birth-process compensators and the conditional birth-rate formula.Source: §A.1 · pp. 64–65
  • Choosing Φ(x) = x² yields the variance identity, while Φ(x) = x log x yields the entropy identity.The entropy specialization uses the stated homogeneity identity with the covariance term.Source: §A.1 · p. 65

A.2 Spectral Independence, Variance, and Entropy

This appendix connects Poisson stochastic localization to variance and entropy evolution through conditional laws and spectral independence. Spectral independence of exponential tilts provides a criterion for entropic stability, which controls entropy evolution.

A.2 Spectral Independence, Variance, and Entropy

  • The conditional law of X1 − Xt given Xt = x is the (1 − t)-thinning of the measure pinned at x.At t = 0, X0 = 0 and the conditional law is µ, yielding the endpoint right-derivative inequality.Source: §A.2 · pp. 65–66
  • Spectral independence of every exponential tilt implies entropic stability, which controls the nonlinear covariance term in the entropy evolution identity.The appendix applies this criterion to conditional laws along localization to obtain an entropy derivative bound.Source: §A.2 · pp. 67–69
  • Along localization, entropy satisfies ∂tE[Entνt[f]] ≥ −η/(1 − t) E[Entνt[f]] for t ∈ (0, 1).The same argument establishes the right-derivative inequality at t = 0.Source: §A.2 · p. 69

A.3 Homogenization for ULC Measures and Lorentzian Polynomials

The section relates ULC measures to Lorentzian polynomials through homogenization, then applies the framework to rapid mixing for the antiferromagnetic q-Potts model.

A.3 Homogenization for ULC Measures and Lorentzian Polynomials

  • For finite downward-closed support, µ is ULC exactly when each sufficiently large homogenized generating polynomial is Lorentzian.The criterion holds for every N at least the maximum ℓ1 norm in the support.Source: §A.3 · p. 69
  • Conversely, Lorentzian homogenization yields the ULC matrix condition, with the remaining boundary cases automatic because no second increment lies in the support.The argument uses the quadratic derivatives’ having at most one positive eigenvalue.Source: §A.3 · p. 70
  • The homogenization argument also establishes that the generating polynomial of a ULC measure is completely log-concave.Complete log-concavity passes from the homogenized polynomials to the generating polynomial by pointwise convergence of the polynomials and their directional derivatives.Source: §A.3 · p. 70

A.4 Rapid Mixing for the Antiferromagnetic q-Potts Model

  • For all sufficiently large Δ, the embedded Potts measure is δ-ULC with δ ≥ 1/2 with probability 1 − o_n(1).Source: §A.4 · p. 72

B Point Processes and Probability Measures on Nd

The section interprets δ-ULC through point-process curvature and relates point-process Glauber dynamics to the canonical birth-death chain. It also presents Poisson stochastic localization as a Poisson bridge construction.

B Point Processes and Probability Measures on Nd

  • The point-process curvature criterion yields a Poincaré inequality, and its finite simple-configuration form is the δ-ULC condition on a hypercube.For the marked point-process lift, the Glauber generator restricted to functions of the count vector agrees with the linear-death generator.Source: B Point Processes and Probability… · pp. 73–74
  • The marked lift uses the factorial weight a(n) = n!µ(n), and its point-process curvature condition transfers the Poincaré inequality to the weighted inequality for µ.The factor n! is the symmetry factor for unlabeled points.Source: B Point Processes and Probability… · pp. 74–75

C Poisson Stochastic Localization and Poisson Bridge

  • The Poisson construction replaces the Brownian score with a multiplicative discrete increment, producing the Poisson–Föllmer process.It arises from a Schrödinger bridge between the point mass at zero and µ under independent rate-one Poisson processes.Source: C Poisson Stochastic Localization and… · p. 76

D Weighted Poincar´e Inequality via Trickle-Down Theorem

The trickle-down proof propagates the local δ-ULC condition backward along Poisson stochastic localization to obtain the weighted Poincaré inequality. The section also characterizes ULC through log-concavity relative to product Poisson measures.

D Weighted Poincar´e Inequality via Trickle-Down Theorem

  • The localization proof propagates local δ-ULC backward independently of the paper’s Bakry–Émery argument.The argument combines a terminal boundary condition with an interior differential inequality, then removes finite-support restrictions by approximation.Source: D Weighted Poincar´e Inequality via… · pp. 77, 80, 81
  • Every δ-ULC measure satisfies the weighted Poincaré inequality with constant δ, equivalently giving the birth-death generator spectral gap at least δ.The trickle-down proof uses the local δ-ULC matrix as the terminal boundary condition for the normalized covariance flow.Source: D Weighted Poincar´e Inequality via… · p. 80
  • For each pinned measure along localization, the thinned law is spectrally independent with parameter 1 / (1 − (1 − δ)(1 − t)).This profile is part of the theorem establishing the weighted Poincaré inequality.Source: D Weighted Poincar´e Inequality via… · p. 80

E Closure Properties of ULC Measures

E.1 Basic Closure Properties

The section establishes closure properties of ULC measures under several common operations. Weighted pinning, products, box truncation, and coordinate conditioning also preserve δ-ULC with the same δ.

E.1 Basic Closure Properties

  • ULC measures remain ULC under exponential tilting, weighted pinning, products, box truncation, and coordinate conditioning.Weighted pinning, products, box truncation, and coordinate conditioning preserve δ-ULC for the same δ.Source: §E.1 · pp. 81–82
  • Weighted pinning, products, box truncation, and coordinate conditioning preserve δ-ULC with the same δ.Source: §E.1 · p. 82

E.2 Marginalization and Coordinatewise Binomial Thinning

ULC measures are closed under marginalization, coordinatewise binomial thinning, and convolution; related test-function formulations show preservation of log-concavity under the Poisson semigroup and binomial convolution.

E.2 Marginalization and Coordinatewise Binomial Thinning

  • Marginalizing an ULC measure onto any coordinate subset yields an ULC measure.The generating-function characterization supplies the closure argument.Source: §E.2 · p. 82
  • Coordinatewise binomial thinning preserves ULC for every p ∈ (0, 1]^d.The equivalent test-function formulation is that the Poisson semigroup preserves log-concavity.Source: §E.2 · pp. 82–83
  • The pure birth semigroup P_t preserves log-concavity whenever P_t f is finite.The stated log-concavity is in the sense of (51).Source: §E.2 · p. 83

E.3 Convolution

  • Convolution of two ULC measures on N^d is ULC.Source: §E.3 · p. 83
  • Binomial convolution preserves log-concavity for functions log-concave in the sense of (51).Source: §E.3 · p. 84

E.4 Particlewise Stochastic Projection

The paper defines a particlewise stochastic projection channel that independently routes or discards each input particle. The channel preserves ULC, encompassing thinning, marginalization, and summing coordinates as special cases.

E.4 Particlewise Stochastic Projection

  • Each input particle is independently assigned to an output type with probability P_ij or discarded with the remaining probability.The output is the vector of counts of the resulting output types.Source: §E.4 · p. 85
  • The channel includes coordinatewise binomial thinning, coordinate marginalization, and summing all input coordinates as special cases.Source: §E.4 · p. 85
  • Particlewise stochastic projection maps a ULC input measure to a ULC output measure.Source: §E.4 · pp. 84–85

Related papers

Loading 2609.23994v1…