When does relative entropy decay exponentially for infinite-dimensional Gibbs samplers?

One L2-to-Lq endpoint bound yields an explicit entropy rate

The paper's central claim is that a spectral gap, by itself, does not control how fast a Gibbs sampler forgets its initial state in relative entropy. A spectral gap gives exponential decay of deviations from equilibrium only in a weighted L2 norm, and in infinite dimension that estimate requires the initial data to lie in the weighted L2 space. Relative entropy D(ρ∥σ) measures distinguishability from the equilibrium Gibbs state σ and, for a Gibbs state, equals the excess free energy divided by the temperature whenever those quantities are finite. Exponential relative-entropy estimates can also apply to initial states outside the weighted L2 space and control convergence in trace distance.

The technical bridge is hyperboundedness: a finite bound on the semigroup as a map from L2(σ) to Lp(σ) for p>2, expressing an improvement in integrability after a finite time. The contractive form, with operator norm at most one, is hypercontractivity. Theorem 1.1 and Theorem 1.2 convert such a bound into exponential relative-entropy decay. In Theorem 1.1, if ∥Ts : L2(σ)→L4(σ)∥ ≤ C for some s>0 and C≥1, and the Stroock–Varopoulos comparison holds, then D(Φt(ρ)∥σ) ≤ e^(−α_HC t) D(ρ∥σ) with α_HC = (s + (log C + 1/2)/λ)^(−1) when C>1, where λ is the spectral gap. If C=1, the same conclusion holds with α_HC = s^(−1) and no gap assumption.

Theorem 1.2 avoids the Stroock–Varopoulos comparison and uses only KMS reversibility. For q0>2 and ∥Ts : L2(σ)→Lq0(σ)∥ ≤ C, the rate is α_KMS = (1−2/q0) / (s + λ^(−1)(log C + 1−2/q0)) when C>1, and α_KMS = (1−2/q0)/s when C=1. The authors note that the finite-dimensional version of this criterion is due to Gao and Wang, and that their rate agrees after converting the normalization D(Φt(ρ)∥σ) ≤ e^(−2α1 t) D(ρ∥σ). The contribution here is extending that criterion to infinite dimension, approximating arbitrary states while preserving the limit of the relative entropy, and proving the required L2-to-Lq bounds for unbounded models.

A positive spectral gap does not imply hypercontractivity

The paper separates two quantities that readers could easily conflate. The spectral gap λ is the optimal exponential contraction rate on mean-zero observables in L2(σ): ∥TtX∥2,σ ≤ e^(−λt)∥X∥2,σ for Tr(σX)=0. Hyperboundedness is a statement about L2-to-Lp smoothing at a finite time. The authors state explicitly that this smoothing property is not implied by a spectral gap alone, and they prove it with a counterexample rather than only asserting it.

Proposition 4.7 gives the general obstruction. For a conservative continuous-time Markov semigroup on a countable set Ω with faithful invariant probability π, if there are points xk with π(xk)→0 and sup_k q(xk) ≤ Λ < ∞, where q(x) is the total rate out of x, then for every p>2 and every finite t>0, ∥Tt : L2(π)→Lp(π)∥ = ∞. The proof takes fk = π(xk)^(−1/2) 1_{xk}, so ∥fk∥2,π = 1, and uses the fact that the probability of no jump before time t starting from xk is at least e^(−Λt). Then ∥Tt fk∥p,π ≥ e^(−Λt) π(xk)^(1/p−1/2) → ∞.

The periodic charging model instantiates this obstruction. On ℓ2(Z) with charge basis (em), charge operator n em = m em, bilateral shift S em = e_{m+1}, Hamiltonian H = h(n) for a nonconstant even polynomial h with positive leading coefficient, and bare jumps S, S†, n, the invariant diagonal sector is a birth–death chain with rates r±_m = |f̂M(h(m±1)−h(m))|^2 bounded between 0 and 1. The invariant probability πm = Z_ch^(−1) e^(−βh(m)) satisfies πm→0 as |m|→∞ while the holding rates satisfy r+_m + r−_m ≤ 2. Proposition 4.7 then shows that the invariant diagonal sector, and hence the full quantum semigroup, has no finite-time L2-to-Lp endpoint for any p>2. The authors nevertheless prove that this sampler has a positive gap, at least min{λ_diag, e^(−1/2)/2} > 0, by analyzing the diagonal and off-diagonal Hilbert–Schmidt sectors separately. The model therefore has a positive gap but no bounded L2-to-Lp map at any finite time, for any p>2.

Full multimode Bose-Hubbard samplers decay exponentially without an occupation cutoff

Theorem 4.3 treats a Bose–Hubbard Hamiltonian on a fixed lattice of k modes, H = Σ_j [ (U_j/2) N_j(N_j − I) − μ_j N_j ] + Σ_{i<j} V_ij N_i N_j − Σ_{i<j} (J_ij a†_i a_j + J_ij a†_j a_i), where a_j and a†_j are mode operators, N_j = a†_j a_j, U_j>0, V_ij≥0, μ_j real, and J_ij complex. With the bare jumps a_j, a†_j, a_j^3, (a†_j)^3 in each mode and the filter f̂M(ν) = exp[−(√(1+(βν)^2) + βν)/4], the sampler has a unique invariant Gibbs state, a positive spectral gap, and a bounded map L2(σβ)→Lp(σβ) for every t>0 and finite p>2. It therefore satisfies D(Φt(ρ)∥σβ) ≤ e^(−αt) D(ρ∥σβ) with α>0.

Two scope conditions matter for reading this result. First, the theorem allows arbitrary hopping amplitudes and uses the full bosonic space, without an occupation cutoff. Second, the cubic jumps are doing specific work: they provide losses of order n^3 on the sector with n particles, which dominate the quadratic energy and the errors caused by hopping. The same theorem gives a bounded endpoint only after a positive waiting time when the smaller jump family a_j, a†_j, a_j^2, (a†_j)^2 is used. The rate may depend on the lattice size and on all model parameters.

The paper also uses this example to make the jump-dependence concrete. For H = N^2 and the filter used in Section 4.1, Theorem 4.2 shows that linear jumps alone have no bounded L2-to-L4 endpoint, whereas adding cubic jumps gives such a bound at every positive time and yields the entropy estimate in Theorem 4.3. Quadratic jumps already give an endpoint after a positive waiting time, including with hopping; cubic jumps give the every-positive-time conclusion. The authors state that the theorem does not give the same entropy estimate for the linear-jump samplers of their earlier work. The behavior depends on the choice of jumps even when the Hamiltonian is fixed.

Trapped Coulomb systems decay exponentially only after finite-rank truncation

Corollary 4.5 considers n particles in d = 2 or 3 dimensions with Hamiltonian H_{n,M} = −Δ + |x|^2 + Π_M ( Σ_{i<j} α_{n,i,j} w_d(x_i − x_j) ) Π_M, where w_2(y) = −log|y|, w_3(y) = |y|^(−1), Π_M = P_M^{⊗n}, and P_M projects onto the first M one-particle Hermite functions. The couplings α_{n,i,j} are real. Only the interaction is truncated; the Hamiltonian acts on L2((R^d)^n). With coordinate creation and annihilation jumps and the Gaussian filter f̂η(ν) = e^(−ην^2 − βν/4), η>0, the sampler has a unique invariant Gibbs state and a positive gap.

For each fixed n and M, the authors prove a contractive L2-to-L4 bound after a finite time and exponential relative-entropy decay. The proof controls the terms containing one ladder operator and one finite-energy coefficient by a weighted oscillator estimate. The constants can depend on n and M. This is the boundary that matters most for readers: the exponential entropy decay is proved for every fixed finite-rank low-energy truncation of the interaction, not for the untruncated Coulomb interaction. The result complements the fixed-cutoff spectral-gap analysis of the authors' earlier work. The form comparison in Section 3 also preserves a tight 2-logarithmic Sobolev inequality, but the authors note that obtaining hypercontractivity from that inequality would require an additional regularity assumption.

Perturbative stability carries hypercontractive estimates to unbounded models

The technical ingredient that makes the applications possible is a perturbative stability theory for hypercontractive estimates and logarithmic Sobolev inequalities, proved in Section 3. Two stability results are stated. First, adding a bounded operator to the generator preserves the Lp bounds along a hypercontractive curve, provided its norm is uniformly bounded on those spaces; the paper gives the resulting norm estimate. Second, a change of the Gibbs weight on a finite-dimensional energy subspace, apart from normalization, preserves a tight 2-logarithmic Sobolev inequality when the Dirichlet forms are comparable and the new generator has a positive gap.

A second structural fact is used repeatedly: in the atomic setting considered here, one bounded L2-to-Lq map already implies compact resolvent and, under primitivity, a positive spectral gap. The argument uses the compactness of the inclusion Lq(σ) ↪ L2(σ) for 2<q<∞. In symmetric Schatten coordinates the inclusion is Jq(Y) = σ^a Y σ^a with a = 1/4 − 1/(2q) > 0, and finite-rank spectral projections Pn ↑ I of σ give ∥Jq − Pn Jq(·) Pn∥_{Tq→T2} ≤ 2 (Tr((I−Pn)σ))^a → 0. Thus a bounded Ts : L2(σ)→Lq(σ) makes e^(−sL) compact, and the spectral theorem then gives compact resolvent for L; if ker L = C√σ, zero is simple and isolated, so gap(L)>0. The authors note that this argument uses the atomic algebra B(H) and trace-class σ, and does not assert compactness for diffuse algebras or a uniform numerical lower bound on the gap.

The value of the gap enters the rate when the endpoint norm exceeds one, which is why the two criteria are stated with separate cases for C>1 and C=1. Corollary 2.1 specializes Theorem 1.2 to the filtered samplers: for σE ∈ (0,∞] and Φt = e^{tL_{σE}} any of the conservative KMS-symmetric semigroups constructed in the authors' earlier work, with invariant Gibbs state σβ and irreducibility giving ker L_{σE} = C√σβ, if ∥T_s^{(2)} : L2(σβ)→L4(σβ)∥ ≤ C > 1, then λ = gap(L) > 0 and the rate is α_KMS = (2s + (2 log C + 1)/λ)^(−1). For C = 1, the rate (2s)^(−1) holds without a gap. This includes the original generator at σE = ∞ and every Gaussian width 0 < σE < ∞; it does not use the Davies limit σE = 0.

The two entropy criteria rest on different assumptions

The paper offers two routes from an endpoint bound to entropy decay, and they are not interchangeable. Theorem 1.1 requires the Stroock–Varopoulos comparison, stated precisely as: for every ρ ∈ D(L) ∩ B(σ), √ρ ∈ D(E) and 4E(√ρ) ≤ EP_L(ρ), where EP_L(ρ) = −Tr(L(ρ)(log ρ − log σ)) is the entropy production. Under the assumptions of Theorem 1.1, the proof also gives the equivalent modified logarithmic Sobolev inequality α_HC D(ρ∥σ) ≤ EP_L(ρ) with the same rate. The authors state that KMS symmetry alone does not imply this comparison, citing Proposition 2.10.

Theorem 1.2 instead obtains entropy contraction directly from KMS symmetry and hypercontractivity, without the Stroock–Varopoulos comparison. The authors explain that the counterexample in Proposition 2.10 shows why the KMS argument is needed for general cross-frequency generators. The two rates are related in the L2-to-L4 case with C>1 and λ>0: α_HC = (s + (log C + 1/2)/λ)^(−1) and α_KMS = α_HC/2.

The comparison can fail for the samplers considered here, which is why the paper does not present a single unified criterion. The authors also note that the Stroock–Varopoulos comparison is one of two arguments used, and that the other uses only the KMS reversibility assumptions. For readers comparing the two theorems, the practical distinction is which hypothesis is easier to verify for a given sampler: the Stroock–Varopoulos comparison for Theorem 1.1, or the KMS reversibility and primitivity assumptions for Theorem 1.2.

Jump order and truncation rank decide whether the rate exists

Across the paper's examples, the decisive variable is not the Hamiltonian alone but the combination of jump family and, for Coulomb systems, truncation rank. For one-mode polynomial number Hamiltonians, Theorem 4.2 compares the degree of the Hamiltonian with the largest jump order, and the outcome is a three-way classification: bounded L2-to-Lp maps occur at every positive time, after a positive waiting time, or never. The stated classification excludes the boundary time in the critical case. The periodic charging model falls in the never category despite having a positive gap.

For Bose–Hubbard, the same logic separates the jump families: cubic jumps a_j, a†_j, a_j^3, (a†_j)^3 give a bounded endpoint at every positive time, while the smaller family a_j, a†_j, a_j^2, (a†_j)^2 gives one only after a positive waiting time. For trapped Coulomb systems, the truncation rank M and particle number n enter the constants, and the result is stated for each fixed n and M. The authors also note that the rate may depend on the lattice size and all model parameters in the Bose–Hubbard case.

The most decision-relevant comparison for a reader is therefore between two samplers that share a Hamiltonian but differ in jumps, or between a truncated and an untruncated interaction. A positive spectral gap is necessary background but not sufficient evidence of exponential relative-entropy decay. The paper's own priority is to supply the additional smoothing estimate, and to show by counterexample what must be added to a spectral-gap estimate. The periodic charging model is the sharpest illustration: it has a positive gap, yet no finite-time L2-to-Lp bound for any p>2, so no entropy-decay rate of the form proved here can be extracted from its gap alone.

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