Disordered systems and (subcritical) polynomial chaos with heavy-tail disorder

Abstract: We study discrete statistical mechanics systems perturbed by a random environment without a finite second moment. Specifically, we consider a random environment whose tail distribution satisfies $P[ω&gt; x] \sim x<sup>{-γ}$ as $x \to +\infty$ for some $γ\in (1,2)$. Inspired by the seminal work of Caravenna, Sun and Zygouras \cite{csz_2016}, we adopt a general framework that encompasses as key examples both the disordered pinning model and the long-range directed polymer model. We provide some subcriticality condition under which we prove that the discrete disordered system possesses a non-trivial scaling limit. We also interpret the subcriticality condition in terms of a generalized Harris criterion without second moment, which gives a prediction for disorder relevance depending on the parameters of the system. Our analysis relies on the study of multilinear polynomials of independent heavy-tailed random variables known as polynomial chaos and their continuous analogue, given by multiple integrals with respect to a $γ$-stable Lévy white noise. We develop precise and flexible moments estimates adapted to the heavy-tailed setting.

• The paper develops an L^p polynomial-chaos framework for 1<γ<2 heavy-tailed disorder, replacing unavailable L^2 methods with decoupling estimates and stable Lévy noise.
• It proves non-degenerate intermediate-disorder limits for pinning when α>1−1/γ and long-range polymers when γ<1+α/d, establishing universal continuum partition functions and Gibbs measures.
• The results rigorously validate the generalized Harris criterion that disorder is relevant when ν<γ/(γ−1), while identifying open challenges for γ≤1 and functional flow convergence.

Overview and motivation

This paper develops a general theory for the intermediate-disorder scaling limits of lattice statistical mechanics models perturbed by an environment whose distribution has a heavy tail. The disorder variables $\omega$ are assumed centered, with a power-law decay of the tail,

$$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$

so that the second moment diverges while the mean remains finite. The framework is inspired by the seminal work of Caravenna, Sun and Zygouras on the Gaussian case [csz_2016], but departs from it in a fundamental way: since $\mathbb{L}^2$ computations are unavailable in the heavy-tail setting, the entire analysis must be rebuilt using $\mathds{L}^p$ estimates for $p \in (1,2)$, based on polynomial chaos expansions of independent heavy-tailed random variables and their continuous counterparts given by multiple stochastic integrals against a $\gamma$-stable Lévy white noise.

The general framework encompasses two canonical examples: the disordered pinning model on a defect line, where the homogeneous reference model is a renewal process with inter-arrival law $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$, $\alpha \in (0,1)$; and the long-range directed polymer model, where the reference model is a random walk in the domain of attraction of an $\alpha$-stable law. The paper proves non-trivial scaling limits under explicit subcriticality conditions, interprets these conditions as a generalized Harris criterion without second moment, and thereby resolves conjectures posed by Berger and Lacoin [berger_lacoin_2021, berger_lacoin_2022] for both models.

Framework: disordered systems as polynomial chaos

The disordered system is defined via a product-form Gibbs modification of the homogeneous measure:

$$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$

The author argues that the product form is not merely a technical convenience relative to the exponential form $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$0: when the disorder is heavy-tailed, the exponential form gives overwhelming weight to extreme values, and $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$1 has no scaling limit after rescaling, whereas $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$2 does. The product form also appears naturally in the connection to SPDEs.

Expanding the partition function yields a multilinear expansion in which the $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$3-point correlation functions of the homogeneous system appear:

$$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$4

where $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$5 combines the disorder intensity, the natural noise scale $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$6 (defined so that $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$7, the cell volume), and the correlation rescaling factor $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$8. The discrete noise $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$9 converges to a $\mathbb{L}^2$0-stable Lévy white noise $\mathbb{L}^2$1, constructed as an almost sure limit of truncated noises in local negative Sobolev spaces $\mathbb{L}^2$2, $\mathbb{L}^2$3. This identifies Lévy noises as having the same Sobolev regularity as the Dirac distribution.

Subcriticality and the generalized Harris criterion

In the finite-variance ($\mathbb{L}^2$4) case, subcriticality corresponds to the condition $\mathbb{L}^2$5, where $\mathbb{L}^2$6 is the correlation exponent and $\mathbb{L}^2$7 the effective dimension of the discretization — precisely Harris' classical prediction that disorder is relevant iff $\mathbb{L}^2$8 [harris_1974]. In the heavy-tail case the analogous dichotomy becomes

$\mathbb{L}^2$9

This criterion had been established only in special cases — the pinning model [lacoin_sohier_2017], and the directed polymer with simple random walk [viveros_2021]. A central contribution of the paper is to put this generalized Harris criterion on rigorous footing in a broad context, covering both guideline models simultaneously.

For the pinning model, the effective dimension is $\mathds{L}^p$0, $\mathds{L}^p$1, and subcriticality reads $\mathds{L}^p$2, with the intermediate window

$\mathds{L}^p$3

where $\mathds{L}^p$4 is the renewal mass function. For the long-range directed polymer, $\mathds{L}^p$5, $\mathds{L}^p$6, and subcriticality reads $\mathds{L}^p$7, with window

$\mathds{L}^p$8

In both cases the limiting object depends only on $\mathds{L}^p$9 through these inequalities, and is universal: it is insensitive to the slowly-varying functions in the renewal law and in the disorder tail, and to the specific discretization. Only the microscopic rescaling of $p \in (1,2)$0 depends on these details.

Main results

The paper establishes three layers of results.

Well-posedness of continuous chaos. Under the summability condition that there exists $p \in (1,2)$1 such that for every $p \in (1,2)$2,

$p \in (1,2)$3

with $p \in (1,2)$4 the symmetric $p \in (1,2)$5 norm (including a factorial $p \in (1,2)$6, mirroring Malliavin calculus conventions), the continuum partition function

$p \in (1,2)$7

is well-defined, non-degenerate, and belongs to $p \in (1,2)$8 for all $p \in (1,2)$9, with the moment bound

$\gamma$0

The construction proceeds via truncation: $\gamma$1 built from the noise truncated at scale $\gamma$2 forms a time-reversed martingale with respect to the filtration generated by large jumps, uniformly integrable in $\gamma$3 for $\gamma$4.

Scaling limit of discrete chaos. If each discrete correlation function converges in symmetric $\gamma$5 norm to a continuous limit, and the family satisfies uniform truncation bounds (finite limsup of $\gamma$6 plus tails vanishing as $\gamma$7), then joint convergence in distribution holds in $\gamma$8:

$\gamma$9

Convergence of Gibbs measures. Provided the homogeneous model converges in distribution on a Polish state space $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$0, the $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$1-weighted correlation functions converge for every bounded continuous test function $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$2, and the continuum partition function is almost surely strictly positive, the full disordered Gibbs measures converge jointly with the noise to a continuum disordered measure. Notably, the proof of tightness in $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$3 does not require boundedness of $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$4, which permits application to the long-range polymer.

Applying this machinery yields the following concrete results:

Model	Subcriticality	Intermediate window	Limit
Pinning	$P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$5	$P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$6	Continuum pinning with $P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$7-stable noise
Long-range polymer	$P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$8	$P(\tau_1=n) \sim c_0 n^{-(1+\alpha)}$9	Continuum polymer on càdlàg paths

Both results establish conjectures stated in [berger_lacoin_2022, §2.5.3] and [berger_lacoin_2021, §2.2]; the polymer result generalizes the $\alpha \in (0,1)$0 case treated previously. Disorder relevance at the level of the partition function is also settled: if $\alpha \in (0,1)$1 then $\alpha \in (0,1)$2 (weak disorder), while if $\alpha \in (0,1)$3 then $\alpha \in (0,1)$4 in probability (strong disorder), under a mild assumption that the normalized occupation field converges to a strictly positive random variable.

Technical core: $\alpha \in (0,1)$5 estimates and decoupling

The moment estimates rest on an $\alpha \in (0,1)$6 decoupling inequality with a remarkable feature: the dependence in the chaos order $\alpha \in (0,1)$7 is linear, $\alpha \in (0,1)$8, with a universal constant $\alpha \in (0,1)$9. This contrasts sharply with general U-statistics decoupling, where one typically obtains $\alpha$0 dependence [delapena_montgomery_1995]. The proof combines Burkholder–Davis–Gundy inequalities with Hitczenko's comparison theorem for tangent processes [hitczenko_1988] — the latter being valid even for $\alpha$1, where convexity fails and the constant can be taken equal to 6.

The key structural idea is a big-jump/small-jump decomposition: each $\alpha$2 is split into a part above the threshold $\alpha$3 and its centered complement. Expanding the product over subsets $\alpha$4, $\alpha$5 estimates are used for big jumps and $\alpha$6 estimates for small jumps, exploiting the asymptotics

$\alpha$7

for $\alpha$8. The resulting bound requires "room" around both endpoints of $\alpha$9: the cases $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$0 and $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$1 would require logarithmic corrections, which is why $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$2 strictly is essential throughout.

The convergence proof for fixed-order chaos uses a two-step approximation: first reduce to smooth compactly supported kernels, then truncate the noise at scale $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$3 and pass to the limit $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$4 using the uniform approximability lemma, which controls $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$5 by choosing $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$6 close enough to $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$7.

Extensions and relation to SPDEs

Beyond stable noises, the paper states a version of the well-posedness theorem for general Lévy white noises, requiring finiteness of the truncated moments $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$8 and $$P_{\Omega_{\delta}}^{\omega,\beta}(d\sigma) := \frac{1}{Z_{\Omega_{\delta}}^{\omega,\beta}} \prod_{x \in \Omega_{\delta}} (1+\beta\omega_x \sigma_x)\; P_{\delta}^{\rm ref}(d\sigma).$$9 for some $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$00. Since general Lévy noises lack scale invariance, they cannot arise as limits of i.i.d. disorders with fixed law; the author circumvents this by allowing $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$01-dependent disorder laws and formulates a natural conjecture in that direction.

The continuum partition functions constructed here are formal candidate solutions to linear SPDEs with multiplicative Lévy noise. For the pinning model with vanishing pinning parameter, the point-to-point limit solves the stochastic Volterra flow equation

$$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$02

for the long-range polymer, the limit flow corresponds to a stochastic fractional heat equation $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$03, reducing to the SHE when $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$04.

Limitations and open questions

Several restrictions are acknowledged explicitly. The approach does not cover $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$05-stable noises with $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$06: polynomial chaos with respect to such noises has no finite $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$07 moments for any $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$08, and finding a general necessary-and-sufficient condition for analogues of the main theorems in that regime remains open. The sufficient condition $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$09 for $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$10 is known to be non-optimal for individual multiple integrals [surgailis_1985], though the paper's estimates hold for arbitrary (not necessarily positive or monotone) kernels, improving on the assumptions of [berger_lacoin_2021, berger_lacoin_2022].

Two delicate points depend on finer properties of the specific model rather than the general correlation-function framework: strict positivity of the continuum partition function (needed for Gibbs-measure convergence), resolved here for pinning via [faugere_lacoin] and for polymers by adapting [berger_lacoin_2022]; and strong disorder beyond the sufficient condition of Theorem on disorder relevance. Finally, the convergence of point-to-point partition function flows is obtained only in the sense of finite-dimensional distributions; strengthening this to convergence in a functional space (càdlàg in time for both models, and spatially regular in the polymer case, particularly in view of intermittency) is left open.

Conclusion

This paper provides a unified treatment of intermediate-disorder scaling limits for disordered systems driven by environments in the domain of attraction of a $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$11-stable law, $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$12. By replacing the unavailable $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$13 machinery with flexible $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$14 estimates built on decoupling inequalities with linear-in-$$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$15 constants, it constructs non-degenerate continuum partition functions and Gibbs measures as multiple integrals against $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$16-stable Lévy white noise, and rigorously validates the generalized Harris criterion $$P[\omega > t] \sim C_0\, t^{-\gamma}, \qquad t \to +\infty,\qquad \gamma \in (1,2),$$17 for disorder relevance across a broad class of models. The applications settle open conjectures for the disordered pinning model and the long-range directed polymer in full generality over their respective subcritical regimes.