Smoothing Inequality for Pinning Models

• The paper establishes a quadratic upper bound on the quenched free energy near criticality, demonstrating that f(h_c+δ) is at most cδ² for small δ.
• It leverages minimal exponential moment conditions on the disorder and uses an interpolation technique to convert tilt smoothing into shift smoothing.
• The result implies the absence of a first-order transition by forcing the critical contact density to zero and ensuring strict convexity of the large-deviation rate function.

Searching arXiv for the cited pinning-model smoothing papers to ground the article in current records. The smoothing inequality for pinning models is a quadratic upper bound on the quenched free energy near criticality that formalizes the rounding of a localization–delocalization transition by disorder. In the disordered renewal pinning model studied in "Disordered pinning models with contact number constraint" (Giacomin et al., 14 Jul 2025), the bound states that, if the quenched critical point $h_c$ is finite and $\mathrm{Var}(\omega_0)>0$, then there exists $c>0$ such that for all sufficiently small $\delta>0$,

$f(h_c+\delta)\le c\delta^2.$

The result is proved under minimal integrability assumptions on the disorder, namely $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$ for some $\eta>0$, $\mathbb{E}[\omega_0]=0$, and $\mathrm{Var}(\omega_0)>0$ (Giacomin et al., 14 Jul 2025). In this framework, the inequality is a structural statement about the critical regularity of the free energy, with consequences for the order parameter, the large-deviation rate function of the contact density, and the geometry of conditioned polymer configurations. A related smoothing mechanism also appears in the Random Walk Pinning Model, where disorder is Markovian rather than i.i.d.; there the quenched free energy satisfies an explicit quadratic bound in dimensions $d\ge 3$ (Berger et al., 2010).

1. Renewal pinning framework and quenched free energy

In the renewal formulation of the disordered pinning model, the polymer is built from a discrete renewal process with i.i.d. inter-arrival times $\mathrm{Var}(\omega_0)>0$0 taking values in $\mathrm{Var}(\omega_0)>0$1 and distributed according to

$\mathrm{Var}(\omega_0)>0$2

where $\mathrm{Var}(\omega_0)>0$3, $\mathrm{Var}(\omega_0)>0$4 is slowly varying at infinity, and $\mathrm{Var}(\omega_0)>0$5 for all $\mathrm{Var}(\omega_0)>0$6 (Giacomin et al., 14 Jul 2025). The renewal epochs are $\mathrm{Var}(\omega_0)>0$7 and $\mathrm{Var}(\omega_0)>0$8, and the corresponding contact indicators are $\mathrm{Var}(\omega_0)>0$9. The total number of contacts up to time $c>0$0 is

$c>0$1

Disorder is introduced through i.i.d. real-valued charges $c>0$2 with mean zero, positive variance, and an exponential moment in a neighborhood of the origin (Giacomin et al., 14 Jul 2025). With average pinning parameter $c>0$3, the finite-volume quenched Gibbs measure is defined by

$c>0$4

where the partition function is

$c>0$5

The quenched free energy is

$c>0$6

Under the stated assumptions, $c>0$7 exists, is finite for all $c>0$8, and is convex, non-decreasing, and Lipschitz with constant $c>0$9 (Giacomin et al., 14 Jul 2025). A concentration inequality gives sub-Gaussian tails for $\delta>0$0 and identifies the same limit almost surely for typical environments (Giacomin et al., 14 Jul 2025).

The critical point is

$\delta>0$1

Then $\delta>0$2 for $\delta>0$3 and $\delta>0$4 for $\delta>0$5, provided $\delta>0$6 (Giacomin et al., 14 Jul 2025). In the localized phase $\delta>0$7, the free energy is strictly convex and infinitely differentiable, and the contact density is

$\delta>0$8

The localized-phase variance $\delta>0$9 controls the limiting scaled variance of $f(h_c+\delta)\le c\delta^2.$0 (Giacomin et al., 14 Jul 2025).

2. Precise form of the smoothing inequality

The central statement is Theorem 1.7 of (Giacomin et al., 14 Jul 2025): if $f(h_c+\delta)\le c\delta^2.$1 and $f(h_c+\delta)\le c\delta^2.$2, then there exists $f(h_c+\delta)\le c\delta^2.$3 such that, for all sufficiently small $f(h_c+\delta)\le c\delta^2.$4,

$f(h_c+\delta)\le c\delta^2.$5

The exponent is exactly $f(h_c+\delta)\le c\delta^2.$6 (Giacomin et al., 14 Jul 2025). In the notation of that work, the constant may be taken explicitly as

$f(h_c+\delta)\le c\delta^2.$7

where $f(h_c+\delta)\le c\delta^2.$8 is any constant satisfying

$f(h_c+\delta)\le c\delta^2.$9

The admissible range of $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$0 is determined by the disorder integrability radius $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$1 and by interpolation constants arising in the proof (Giacomin et al., 14 Jul 2025).

The inequality is entirely quenched. It is stated for the quenched free energy $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$2 and does not rely on an annealed estimate (Giacomin et al., 14 Jul 2025). The disorder assumptions are deliberately minimal: finiteness of the log-moment generating function

$\mathbb{E}[e^{\eta |\omega_0|}]<\infty$3

for $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$4, together with $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$5 and $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$6 (Giacomin et al., 14 Jul 2025). No boundedness assumption on $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$7 is imposed, and no special tail conditions beyond a local exponential moment are required. The renewal tail exponent $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$8 enters the bound through the coefficient $\mathbb{E}[e^{\eta |\omega_0|}]<\infty$9 inherited from a tilt-based smoothing estimate due to Caravenna–den Hollander, and through technical renewal estimates (Giacomin et al., 14 Jul 2025).

A key structural implication is that the right derivative at criticality vanishes. Since $\eta>0$0, one has

$\eta>0$1

hence the critical contact density

$\eta>0$2

is equal to $\eta>0$3 in the disordered model (Giacomin et al., 14 Jul 2025). This excludes a first-order transition in the sense of a jump of the order parameter.

3. Mechanism of the proof: from tilt smoothing to shift smoothing

The proof in (Giacomin et al., 14 Jul 2025) proceeds by comparing two perturbations of the system: tilting the disorder distribution and shifting the pinning parameter. On blocks of size $\eta>0$4, the tilted disorder measure is

$\eta>0$5

and the associated tilted free energy is defined as

$\eta>0$6

Caravenna–den Hollander’s tilted smoothing estimate gives

$\eta>0$7

for all $\eta>0$8, where $\eta>0$9 depends on the moment-generating-function radius of the disorder (Giacomin et al., 14 Jul 2025).

The new ingredient is an interpolation inequality that shows tilt can be dominated by an appropriate shift of $\mathbb{E}[\omega_0]=0$0. There exist $\mathbb{E}[\omega_0]=0$1 and $\mathbb{E}[\omega_0]=0$2 such that, for all $\mathbb{E}[\omega_0]=0$3, $\mathbb{E}[\omega_0]=0$4, $\mathbb{E}[\omega_0]=0$5, and any site $\mathbb{E}[\omega_0]=0$6,

$\mathbb{E}[\omega_0]=0$7

where

$\mathbb{E}[\omega_0]=0$8

is the tilt mean (Giacomin et al., 14 Jul 2025). Integrating in $\mathbb{E}[\omega_0]=0$9 from $\mathrm{Var}(\omega_0)>0$0 to $\mathrm{Var}(\omega_0)>0$1 yields

$\mathrm{Var}(\omega_0)>0$2

for all small $\mathrm{Var}(\omega_0)>0$3 (Giacomin et al., 14 Jul 2025). Substituting $\mathrm{Var}(\omega_0)>0$4 and combining with the tilt-smoothing bound gives

$\mathrm{Var}(\omega_0)>0$5

with $\mathrm{Var}(\omega_0)>0$6.

At the technical level, the comparison relies on local changes of the environment at one site and the binary nature $\mathrm{Var}(\omega_0)>0$7 (Giacomin et al., 14 Jul 2025). Convexity estimates for $\mathrm{Var}(\omega_0)>0$8 under the replacement $\mathrm{Var}(\omega_0)>0$9 lead to inequalities of the form

$d\ge 3$0

and also

$d\ge 3$1

A crucial lower bound is

$d\ge 3$2

for small $d\ge 3$3, ensured by $d\ge 3$4 and dominated convergence (Giacomin et al., 14 Jul 2025). This identifies an admissible $d\ge 3$5 and closes the interpolation argument.

The interpolation itself is expressed through the derivative

$d\ge 3$6

and its integration over $d\ge 3$7 yields the monotonic comparison between shift and tilt (Giacomin et al., 14 Jul 2025). This is the core of the smoothing mechanism under minimal integrability.

4. Relation to earlier smoothing results

The smoothing inequality in (Giacomin et al., 14 Jul 2025) extends prior results in the disordered pinning literature. Giacomin–Toninelli proved smoothing for depinning and pinning transitions under bounded disorder, using coarse graining and entropy bounds; boundedness of $d\ge 3$8 was a substantive assumption in that approach (Giacomin et al., 14 Jul 2025). Caravenna–den Hollander later established a general smoothing inequality via disorder tilting for charges of the form $d\ge 3$9, assuming finite moment generating function on a neighborhood $\mathrm{Var}(\omega_0)>0$00 and the constraint $\mathrm{Var}(\omega_0)>0$01; their Theorem 1.5 gives

$\mathrm{Var}(\omega_0)>0$02

for small $\mathrm{Var}(\omega_0)>0$03 (Giacomin et al., 14 Jul 2025).

The contribution of (Giacomin et al., 14 Jul 2025) is to convert this tilt smoothing into a shift smoothing statement at criticality under the weaker assumption $\mathrm{Var}(\omega_0)>0$04 for some $\mathrm{Var}(\omega_0)>0$05, without boundedness and without stronger tail control. The article explicitly notes that this covers regimes in which $\mathrm{Var}(\omega_0)>0$06, which are not captured by the $\mathrm{Var}(\omega_0)>0$07 constraint when that condition is translated into the normalization used there (Giacomin et al., 14 Jul 2025). The exponent $\mathrm{Var}(\omega_0)>0$08 is preserved, and the constant is made explicit through $\mathrm{Var}(\omega_0)>0$09 and $\mathrm{Var}(\omega_0)>0$10 (Giacomin et al., 14 Jul 2025).

A distinct but related instance of smoothing arises in the Random Walk Pinning Model. In "The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics" (Berger et al., 2010), for all $\mathrm{Var}(\omega_0)>0$11, $\mathrm{Var}(\omega_0)>0$12, and $\mathrm{Var}(\omega_0)>0$13, the quenched free energy satisfies

$\mathrm{Var}(\omega_0)>0$14

where $\mathrm{Var}(\omega_0)>0$15 (Berger et al., 2010). There the disorder is generated by a random walk $\mathrm{Var}(\omega_0)>0$16, and smoothing is proved by increasing the jump rate from $\mathrm{Var}(\omega_0)>0$17 to $\mathrm{Var}(\omega_0)>0$18, controlling the Radon–Nikodym cost by Poisson large deviations, and combining coarse graining with superadditivity (Berger et al., 2010). This provides an explicit quadratic upper bound in a Markovian environment and shows that the smoothing paradigm extends beyond i.i.d. disorder.

5. Consequences for phase structure and rate functions

The principal phase implication of the smoothing inequality is the disappearance of a first-order transition in the disordered model. In the pure renewal pinning model with finite mean inter-arrival time, that is $\mathrm{Var}(\omega_0)>0$19, the transition at $\mathrm{Var}(\omega_0)>0$20 is first-order:

$\mathrm{Var}(\omega_0)>0$21

(Giacomin et al., 14 Jul 2025). By contrast, in the disordered model the quadratic bound forces $\mathrm{Var}(\omega_0)>0$22, so

$\mathrm{Var}(\omega_0)>0$23

and the contact fraction has no discontinuity at criticality (Giacomin et al., 14 Jul 2025). The result therefore shows that disorder rounds the transition.

This rounding has direct consequences for the large-deviation rate function of the empirical contact density. In (Giacomin et al., 14 Jul 2025), the rate function is defined by the Legendre–Fenchel transform

$\mathrm{Var}(\omega_0)>0$24

An explicit representation is given:

$\mathrm{Var}(\omega_0)>0$25

$\mathrm{Var}(\omega_0)>0$26

and

$\mathrm{Var}(\omega_0)>0$27

where $\mathrm{Var}(\omega_0)>0$28 denotes the inverse of $\mathrm{Var}(\omega_0)>0$29 (Giacomin et al., 14 Jul 2025). Since smoothing implies $\mathrm{Var}(\omega_0)>0$30, the affine stretch on $\mathrm{Var}(\omega_0)>0$31 disappears, and $\mathrm{Var}(\omega_0)>0$32 becomes strictly convex on $\mathrm{Var}(\omega_0)>0$33 and of class Gevrey-3 (Giacomin et al., 14 Jul 2025).

This strict convexity is central to the paper’s broader analysis of constrained pinning. It excludes the flat portions in the rate function associated with phase coexistence and supports a genuinely localized interpretation of configurations conditioned to have a prescribed positive but atypical contact density (Giacomin et al., 14 Jul 2025). The article explicitly connects the smoothing inequality to this convexity mechanism.

The corresponding order-parameter statement in the Random Walk Pinning Model has the same qualitative content. Since the free energy obeys a quadratic bound above $\mathrm{Var}(\omega_0)>0$34, the right derivative at criticality is zero, and the order parameter

$\mathrm{Var}(\omega_0)>0$35

has no jump at $\mathrm{Var}(\omega_0)>0$36 (Berger et al., 2010). Corollary 1.7 of (Berger et al., 2010) further shows that at criticality the intersection local time is subdiffusive in the sense that

$\mathrm{Var}(\omega_0)>0$37

in probability for every $\mathrm{Var}(\omega_0)>0$38.

6. Role within conditioned localization and the quenched local CLT

In (Giacomin et al., 14 Jul 2025), the smoothing inequality is not an isolated regularity statement; it is a prerequisite for the paper’s analysis of the model under a contact-number constraint. The underlying problem concerns the geometry of polymer configurations conditioned on $\mathrm{Var}(\omega_0)>0$39 with $\mathrm{Var}(\omega_0)>0$40 in a prescribed positive-density regime. Without disorder, when the pure transition is discontinuous and the density is constrained to lie below the minimum typical localized density, the system exhibits a big jump phenomenon: one macroscopic gap is created between two contacts, while the remainder of the path retains localized characteristics (Giacomin et al., 14 Jul 2025). The paper emphasizes that, in the presence of bounded disorder, this phenomenon is no longer observed and the largest gap in the conditioned system is $\mathrm{Var}(\omega_0)>0$41 (Giacomin et al., 14 Jul 2025).

The main result of (Giacomin et al., 14 Jul 2025) strengthens this substantially. Under minimal integrability assumptions on the disorder, the conditioned system is localized in a very strong sense, and in particular the largest gap is $\mathrm{Var}(\omega_0)>0$42 (Giacomin et al., 14 Jul 2025). The proof combines two ingredients: smoothing, which yields $\mathrm{Var}(\omega_0)>0$43 and strict convexity of the large-deviation rate, and a quenched local central limit theorem for $\mathrm{Var}(\omega_0)>0$44 in the localized phase, which provides sharp control of the mass $\mathrm{Var}(\omega_0)>0$45 (Giacomin et al., 14 Jul 2025).

The article is explicit that the smoothing inequality itself does not require the local CLT, but the CLT is used downstream for conditional statements (Giacomin et al., 14 Jul 2025). The local Gaussian control around $\mathrm{Var}(\omega_0)>0$46, together with the strict convexity coming from smoothing, yields the conditional localization theorems: if $\mathrm{Var}(\omega_0)>0$47 stays in a closed subset of $\mathrm{Var}(\omega_0)>0$48, then the largest gap $\mathrm{Var}(\omega_0)>0$49 is $\mathrm{Var}(\omega_0)>0$50 and mesoscopic averages of the contact indicators $\mathrm{Var}(\omega_0)>0$51 remain close to $\mathrm{Var}(\omega_0)>0$52 uniformly down to scales diverging faster than $\mathrm{Var}(\omega_0)>0$53 (Giacomin et al., 14 Jul 2025). This suggests that smoothing is not merely a statement about critical differentiability; it governs the admissible geometry of atypical localized states.

A common misconception is to interpret smoothing only as the absence of a discontinuity in the first derivative of the free energy. In the setting of (Giacomin et al., 14 Jul 2025), its role is broader: the vanishing of $\mathrm{Var}(\omega_0)>0$54 eliminates affine parts in the rate function, which in turn rules out coexistence scenarios compatible with a macroscopic gap under contact-density conditioning. The paper’s conditional results rely precisely on this chain of implications.

7. Scope, optimality, and conceptual significance

Within the regime $\mathrm{Var}(\omega_0)>0$55, the smoothing exponent is independent of the tail parameter $\mathrm{Var}(\omega_0)>0$56 (Giacomin et al., 14 Jul 2025). The quantity $\mathrm{Var}(\omega_0)>0$57 enters the constant, through the factor $\mathrm{Var}(\omega_0)>0$58, but not the power of $\mathrm{Var}(\omega_0)>0$59. The paper presents this as consistent with the general “rounding by randomness” paradigm associated with Imry–Ma reasoning: disorder removes first-order behavior and rounds the transition to at least second order (Giacomin et al., 14 Jul 2025). In this sense, the exponent $\mathrm{Var}(\omega_0)>0$60 is optimal as a smoothing exponent for first-order transitions produced by quenched randomness, although matching lower bounds

$\mathrm{Var}(\omega_0)>0$61

are not established there (Giacomin et al., 14 Jul 2025).

The assumptions of the result are deliberately close to minimal. The disorder need not be bounded, and no heavy technical tail conditions beyond a local exponential moment are required (Giacomin et al., 14 Jul 2025). This sharply separates the new theorem from earlier approaches that relied on bounded disorder or stronger integrability windows. The use of the interpolation inequality to convert tilt smoothing into shift smoothing is the key novelty at the methodological level (Giacomin et al., 14 Jul 2025).

The broader conceptual significance is visible by comparison with other pinning settings. In the Random Walk Pinning Model, the same quadratic rounding phenomenon survives in a non-i.i.d. environment, with an explicit constant depending on the dimension, the jump-rate parameter, and the Green function $\mathrm{Var}(\omega_0)>0$62 (Berger et al., 2010). There the contrast with the annealed model is particularly sharp: for $\mathrm{Var}(\omega_0)>0$63, the annealed transition is first-order, while the quenched transition is at least second-order (Berger et al., 2010). A plausible implication is that quadratic smoothing is a robust signature of quenched disorder across several pinning-type systems, even though the mechanism implementing the effective “tilt” depends strongly on the environment structure.

Taken together, these results place the smoothing inequality at the center of the modern theory of disordered pinning. In the renewal setting, it establishes that

$\mathrm{Var}(\omega_0)>0$64

under minimal integrability of the disorder, implies $\mathrm{Var}(\omega_0)>0$65, forces strict convexity of the contact-density large-deviation rate on $\mathrm{Var}(\omega_0)>0$66, and underlies sharp logarithmic-gap localization under contact-number constraints (Giacomin et al., 14 Jul 2025). In related Markovian pinning models, it likewise expresses the rounding of the critical point by disorder through an explicit quadratic bound (Berger et al., 2010).