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Pinning models with contact number constraint: the effect of disorder

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Pith's one-line read Disordered pinning models with non-degenerate random charges have no big-jump regime: conditioning the contact number to any density in $(0,1)$ leaves the largest gap at most logarithmic in system size.

desk verdict Genuine O(log n) strengthening of the no-big-jump result, but acceptance should be conditional on the companion preprint [34] checking out. read the letter →

arxiv 2507.10707 v2 pith:HZYEBFI7 submitted 2025-07-14 math.PR cond-mat.stat-mechmath-phmath.MP

Giambattista Giacomin , Marco Zamparo This is my paper · ORCID
classification math.PRcond-mat.stat-mechmath-phmath.MP MSC 60K3782B4460K3560F05
keywords disorderedpinningmodelsbigjumpphenomenondisordersmoothingcontactnumberconstraintquenchedlocalcentrallimittheoremlargedeviationsrenewalprocessesdensity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In a disordered pinning model, renewal epochs are contacts that pay an energy $h+\omega_a$, with the charges $\omega_a$ independent and identically distributed. In the pure (no-disorder) model with a first-order localization transition, conditioning the contact density to lie below the localized minimum forces one macroscopic gap between contacts: the big jump phenomenon. This paper proves that adding disorder with non-zero variance destroys that phenomenon: for typical disorder, whenever the contact number is conditioned to a density in a closed subinterval of $(0,1)$, the largest gap $M_n$ between contacts is at most $O(\log n)$, with probability tending to $1$ uniformly in the conditioning level. The mechanism is that disorder smooths the transition, making the critical contact density $\rho_c$ equal to zero, and a new quenched local central limit theorem controls the probability of each contact number. This matters because the big jump is the one-dimensional analogue of phase-separation droplets, so the result says quenched disorder erases that droplet mechanism entirely.

What carries the argument

The central object is the maximal gap $M_n=\max\{T_1,\dots,T_{L_n}\}$ under the sharp conditioning $L_n=l$. The argument changes measure to the unique $h_{n,l,\omega}$ with $E_{n,h_{n,l,\omega},\omega}[L_n]=l$, so the conditioned probability factors into a numerator $P_{n,h,\omega}(M_n>c\log n)$ and a denominator $P_{n,h,\omega}(L_n=l)$. The numerator is controlled by Lemma 3.7, which uses the alternative free energy $\mu(h)$, the exponential decay rate of the probability of a renewal at time $n$, to show $P_{n,h,\omega}(M_n>c\log n)$ decays faster than any power of $n$. The denominator is controlled by the quenched local central limit theorem of Theorem 1.9, proved by bounding the characteristic function of $L_n$ through a sparse observable $J_{n,h,\omega}$ built on even sites between odd contacts and using exponential two-replica decorrelation. The disorder-smoothing Theorem 1.5 enters by guaranteeing $\rho_c=0$, so the inverse map of the contact density covers all of $(0,1)$ and the tilted parameter $h_{n,l,\omega}$ stays in a compact localized set uniformly in $l$.

What would settle it

Run a simulation of a disordered pinning model with i.i.d. charges of small non-zero variance and a heavy-tailed renewal law, condition on $L_n=l$ with $l/n$ fixed in $(0,1)$, and estimate $P_{n,h,\omega}(M_n>c\log n\mid L_n=l)$ as $n$ grows; if for some fixed $c$ this probability fails to tend to zero, Theorem 1.7 is false. A cheaper internal check is to compute the two-replica overlap sum in Lemma 3.2: if $\mathbb E[\sup_{h\in H}\mathbb E^{\otimes 2}_{n,h,\omega}[\prod_{k=1}^{n-1}(1-X_kX'_k)]]$ does not decay exponentially in $n$, the decorrelation input fails and the proof collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.7: under Assumptions 1.1 and 1.2 with $\int \omega_0^2\,d\mathbb P>0$, for $\mathbb P$-a.e. disorder realization $\omega$, for every $h\in\mathbb R$ and every closed $R\subset(0,1)$ there is a constant $c>0$ independent of $\omega$ such that $\lim_{n\to\infty}\sup_{l\in nR\cap\mathbb N}P_{n,h,\omega}(M_n>c\log n\mid L_n=l)=0$. In words, conditioning the contact density to be any fixed fraction of the volume leaves the configuration localized at the microscopic scale: no single inter-contact gap can be macroscopic. Theorem 1.8 refines this to mesoscopic windows, showing the local contact density is uniformly close to the conditioned density on every interval longer than $\zeta_n\log n$ with $\zeta_n\to\infty$. The structural reason is Corollary 1.6: disorder forces the rate function $I_h$ for the contact density to be strictly convex on $(0,1)$, eliminating the affine stretch that in the pure model is realized by exactly one large gap. The proof also establishes a quenched local central limit theorem for $L_n$ in the localized phase.

Load-bearing premise

The load-bearing premise is that the localized-phase estimates of the companion preprint [34] are correct as stated, because Theorem 1.5, Lemma 3.6, and Lemma 3.7 import them without independent proof, and the main theorems inherit any error there.

Editorial extensions

If this is right

  • For typical disorder, every closed subinterval conditioning on contact density gives uniform logarithmic gaps: the pure model's big-jump mechanism has no disordered analogue at any $h$.
  • The largest gap in the conditioned model has the same $O(\log n)$ scale as the largest gap in the unconditioned localized phase, so conditioning does not create a new phase.
  • The quenched local central limit theorem supplies a uniform Gaussian approximation of the contact-number mass function in the localized phase, a tool usable beyond conditioning.
  • Soft conditioning inherits the bound: $P_{n,h,\omega}(M_n>c\log n\mid L_n\ge rn)\to0$, and for $h>h_c$ also under $L_n\le rn$; circular-DNA models with nonlocal potentials acquire $M_n\le c\log n$ when the contact density sits in a localized window.
  • The rate function for the contact density loses its affine stretch and becomes strictly convex and Gevrey-3 on $(0,1)$, merging the two large-deviation regimes of the pure model into one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural test of sharpness is to simulate the conditional law for bounded i.i.d. charges and see whether the gap distribution is actually $\Theta(\log n)$, not $o(\log n)$; the paper proves an upper bound and does not claim tightness.
  • The local-CLT route is likely transferable to contact densities approaching $1$, where there is almost no room for a large gap; the paper notes that bounded disorder should suffice to push the argument into that range.
  • The disorder-smoothing result strengthens the analogy with random-field Ising systems: a random field that rounds a first-order transition should destroy macroscopic phase-separation droplets in any model whose free energy is sufficiently regular, an analogy the paper draws but does not prove.
  • For the generalized Poland–Scheraga model built on two-dimensional renewals, it remains open whether disorder washes out the big jump; a numerical check of the conditional gap profile in that model would be a direct extension of Theorem 1.7.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies disordered pinning models with i.i.d. charges satisfying a subexponential moment condition. Its main result (Theorem 1.7) is that, for P-a.e. disorder, conditioning the contact count to lie in nR for a closed R⊂(0,1) forces the maximal gap between contacts to be O(log n) with high probability, uniformly in the conditioning level; this strengthens previous o(n) bounds. The proof combines a new quenched local CLT for the contact number (Theorem 1.9), a smoothing inequality f(h_c+δ)≤cδ² (Theorem 1.5), and a change of measure to a canonical parameter h_{n,l,ω}; Theorem 1.8 states a mesoscopic homogeneity result. Section 4 treats soft conditioning and circular-DNA models. The paper is carefully written and the main derivations are explicit, but several localized-phase inputs are imported from the companion preprint [34].

Significance. If the results hold, this is a substantial advance in disordered pinning: it shows the big-jump regime is completely washed out under minimal integrability, not merely at the macroscopic o(n) scale, and it provides a local CLT and a smoothing inequality that are of independent interest. The proof is parameter-free in the sense that no fitted constants drive the conclusion; the O(log n) bound is derived from explicit estimates. I note that the central claim rests on the companion paper [34] (Sections 3.1 and Lemma 3.7), so the verification risk flagged in the stress-test note is real; however, the dependency is explicit and the imported statements are listed, and I do not see an internal inconsistency.

minor comments (4)
  1. [Section 3.1] Please add a sentence on the status of the companion paper [34] and specify exactly which results are used in the form stated here; the central theorems inherit the correctness of [34], so the reader should know whether it is published, accepted, or available only as a preprint.
  2. [Proof of Lemma 3.9] In the upper-tail Chernoff bound there is a sign typo: the display "e^{−ρ(h)ζn+ϵζn/2}" should read "e^{−ρ(h)ζn−ϵζn/2}" to match the subsequent line and the final estimate (3.20).
  3. [Abstract and Section 1.5] The abstract states the O(log n) result without the restriction to closed subsets of (0,1); please make this restriction explicit in the abstract so that it matches Theorem 1.7 and the limitation discussion in Section 1.6.3.
  4. [Section 3.4] The derivation of Theorem 1.8 from Theorem 3.8 is only a one-sentence reduction; please spell out the change-of-measure argument or give a precise reference to the analogous steps in the proof of Theorem 1.7, since Theorem 1.8 is a headline result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the O(log n) conditioned-gap theorem is proved from independent local-CLT and gap estimates; companion-paper dependence is a verification risk, not circularity.

full rationale

I found no circular step. Theorem 1.7 is derived by the change of measure (3.16) to the unique h_{n,l,omega} with E_{n,h_{n,l,omega},omega}[L_n]=l, the quenched local CLT (Theorem 1.9) for a lower bound on P_{n,h,omega}[L_n=l], and Lemma 3.7 for an upper bound on P_{n,h,omega}[M_n>c log n]; none of these inputs is defined in terms of the target probability, and the conclusion is not an identity with any of them. The local CLT is proved in the paper from Lemmas 3.5 and 3.6 and bound (3.13), using the quenched CLT from [34] only as a standard limiting ingredient. Lemma 3.7 is proved from the alternative free energy mu(h)>0 and Birkhoff's ergodic theorem. The genuinely external load-bearing input is the companion paper [34] (strict convexity and Gevrey-3 regularity of f, two-replica decorrelation Lemmas 3.1-3.2, concentration Theorem 3.3, quenched CLT Theorem 3.4, positivity of mu); this is a self-citation and a verification risk because [34] is by the same authors and is not reproved or machine-checked here. However, it is a stated result set in a separate paper, not a fitted parameter or a definitional reuse of Theorem 1.7, so it does not make the derivation circular. Similarly, the LDP in Proposition 1.2 is quoted from [39] (with overlapping authorship) but is used only in the soft-conditioning corollaries. I see no self-definitional, fitted-input-as-prediction, uniqueness-import, ansatz-smuggling, or renaming step. Hence score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical entities. Auxiliary objects such as the alternative free energy mu(h) and the random variable J_{n,h,omega} are technical devices. The central theorems depend on the renewal/disorder assumptions and on the unproved companion results [34].

assumptions (5)
  • domain assumption Renewal inter-arrival distribution satisfies Assumption 1.1 (p(t)=ell(t)/t^{alpha+1}, alpha>=1, ell slowly varying, p(t)>0).
    Model class; used throughout, in particular for regular variation properties in Lemmas 3.6 and 3.7 and Appendix A.
  • domain assumption Disorder charges are i.i.d., centered, with E[e^{eta|omega0|}]<infinity for some eta>0 (Assumption 1.2).
    Minimal integrability condition; used in the smoothing argument, concentration, and the local CLT.
  • domain assumption Non-degenerate disorder: E[omega0^2]>0.
    Needed for Theorem 1.5 to give rho_c=0 and for the affine stretch to disappear; without it the big jump can survive.
  • domain assumption Companion results of [34] (strict convexity/Gevrey regularity, two-replica estimates, concentration, quenched CLT, alternative free energy positivity).
    Assumed as established background; these are the main external load-bearing inputs.
  • standard math Classical results: Birkhoff ergodic theorem, local CLT for triangular arrays, regular variation theorems of Bingham-Goldie-Teugels.
    Used in Lemmas 3.6, 3.7, 3.9 and Appendix A.

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Pith. "Pith review of Pinning models with contact number constraint: the effect of disorder." pith.science (2026). https://pith.science/paper/HZYEBFI7

@misc{pith2026250710707,
  author       = {Pith},
  title        = {Pith review of: Pinning models with contact number constraint: the effect of disorder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZYEBFI7}},
  note         = {Machine review of arXiv:2507.10707}
}
abstract

Disordered pinning models are statistical mechanics models built on discrete renewal processes: renewal epochs in this context are called contacts. It is well known that pinning models can undergo a localization/delocalization phase transition: in the localized phase the typical density of contacts is positive and the largest gap between contacts is at most of the order of the logarithm of the size $n$ of the system, whereas the system is void of contacts in the delocalized phase. When disorder is absent and the phase transition is discontinuous, conditioning the contact density to be positive but smaller than the minimum typical density in the localized phase has the effect of forcing to create one, and only one, macroscopic gap between two contacts, while the rest of the configuration keeps the characteristics of a localized state. However it is known that, in the presence of (bounded) disorder, this big jump phenomenon is no longer observed, in the sense that the largest gap in the conditioned system is $o(n)$. Under minimal integrability conditions on the disorder we show that the conditioned system is localized in a very strong sense, in particular the largest gap is $O(\log n)$. The proof is achieved by exploiting the improved understanding of the localized phase of disordered pinning models developed in [34] and by establishing some refined estimates, in particular a quenched Local Central Limit Theorem. We also present an analysis of the effect of disorder on the large deviation rate function of the contact density and, as an important ingredient for the generality of our results, we establish a smoothing inequality for pinning models under minimal integrability conditions, thus generalizing [11,32].

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Reference graph

Works this paper leans on

50 extracted references · 49 canonical work pages

  1. [34]

    Concentration and fluctuation phenomena in the localized phase of the pinning model

    G. Giacomin and M. Zamparo, Concentration and fluctuation phenomena in the localized phase of the pinning model, arXiv:2405.16991

    work page Pith review arXiv

  2. [29]

    Giacomin and B

    G. Giacomin and B. Havret, Localization, big-jump regime and the effect of disorder for a class of generalized pinning models, J. Stat. Phys. 181 (2020), 2015–2049

    work page 2020

  3. [1]

    Aizenman, M

    M. Aizenman, M. Harel, and R. Peled, Exponential decay of correlations in the 2D random field Ising model, J. Stat. Phys. 180 (2020), 304–331

    work page 2020

  4. [2]

    Aizenman and J

    M. Aizenman and J. Wehr, Rounding effects of quenched randomness on first-order phase transitions, Comm. Math. Phys. 130 (1990), 489–528

    work page 1990

  5. [3]

    Armend´ ariz and M

    I. Armend´ ariz and M. Loulakis, Conditional distribution of heavy tailed random variables on large deviations of their sum, Stoch. Process. Their Appl. 121 (2011), 1138–1147

    work page 2011

  6. [4]

    A. Bar, A. Kabak¸ cıo˘ glu, and D. Mukamel, Denaturation of circular DNA: Supercoils and overtwist, Phys. Rev. E 86 (2012), 061904

    work page 2012

  7. [5]

    Berger, G

    Q. Berger, G. Giacomin, and M. Khatib, Disorder and denaturation transition in the generalized Poland-Scheraga model, Ann. H. Lebesgue 3 (2020), 299–339

    work page 2020

  8. [6]

    Billingsley, Probability and measure, 3rd edn (John Wiley & Sons, New York, 1995)

    P. Billingsley, Probability and measure, 3rd edn (John Wiley & Sons, New York, 1995)

    work page 1995

Show all 50 references
  1. [7]

    Bingham, C.M

    N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation (Cambridge University Press, Cam- bridge, 1989)

    1989

  2. [8]

    Bodineau, D

    T. Bodineau, D. Ioffe, and Y. Velenik, Rigorous probabilistic analysis of equilibrium crystal shapes, J. Math. Phys. 41 (2000), 1033–1098

    2000

  3. [9]

    Bricmont and A

    J. Bricmont and A. Kupiainen, Phase transition in the 3d random field Ising model, Comm. Math. Phys. 116 (1988), 539–572

    1988

  4. [10]

    Campanino, D

    M. Campanino, D. Capocaccia, and B. Tirozzi, The local central limit theorem for Gibbs random fields, Commun. Math. Phys. 70 (1979), 125–132

    1979

  5. [11]

    Caravenna and F

    F. Caravenna and F. den Hollander, A general smoothing inequality for disordered polymers, Electron. Commun. Probab. 18 (2013), 1–15

    2013

  6. [12]

    Cassandro, I

    M. Cassandro, I. Merola and P. Picco, Phase Separation for the long range one-dimensional Ising Model, J. Stat. Phys. 167 (2017), 351–382

    2017

  7. [13]

    Cassandro, E

    M. Cassandro, E. Orlandi, and P. Picco, Phase transition in the 1d random field Ising model with long range interaction, Comm. Math. Phys. 288 (2009), 731–744

    2009

  8. [14]

    Cerf and ´A

    R. Cerf and ´A. Pisztora, On the Wulff crystal in the Ising model, Ann. Probab. 28 (2000), 947–1017

    2000

  9. [15]

    Davis and D

    B. Davis and D. McDonald, An elementary proof of the local central limit theorem, J. Theor. Probab. 8 (1995), 693–701

    1995

  10. [16]

    Denisov, A.B

    D. Denisov, A.B. Dieker, and V. Shneer, Large deviations for random walks under subexponentiality: the big-jump domain, Ann. Probab. 36 (2008), 1946–1991

    2008

  11. [17]

    J. Ding, F. Huang, and J. Maia, Phase transitions in low-dimensional long-range random field Ising models, arXiv:2412.19281

    arXiv

  12. [18]

    Ding and Z

    J. Ding and Z. Zhuang, Long range order for random field Ising and Potts models, Comm. Pure Appl. Math. 77 (2024), 37–51. DISORDERED PINNING MODELS WITH CONTACT NUMBER CONSTRAINT 33

    2024

  13. [19]

    Ding and J

    J. Ding and J. Xia, Exponential decay of correlations in the two-dimensional random field Ising model, Invent. Math. 224 (2021), 999–1045

    2021

  14. [20]

    Dobrushin, R

    R. Dobrushin, R. Koteck´ y, and S. Shlosman,Wulff construction: a global shape from local interaction, Translations of Mathematical Monographs 104 (American Mathematical Society, Providence, 1992)

    1992

  15. [21]

    Dobrushin and B

    R.L. Dobrushin and B. Tirozzi, The central limit theorem and the problem of equivalence of ensembles, Comm. Math. Phys. 54 (1977), 173–192

    1977

  16. [22]

    Doney, A large deviation local limit theorem, Math

    R.A. Doney, A large deviation local limit theorem, Math. Proc. Camb. Phil. Soc. 105 (1989), 575–577

    1989

  17. [23]

    Ferrari, C

    P.A. Ferrari, C. Landim, and V.V. Sisko, Condensation for a fixed number of independent random variables, J. Stat. Phys. 128 (2007), 1153–1158

    2007

  18. [24]

    S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distri- butions, (Springer, New York, 2011)

    2011

  19. [25]

    Garel and H

    T. Garel and H. Orland, Generalized Poland-Scheraga model for DNA hybridization , Biopolymers 75 (2004), 453–467

    2004

  20. [26]

    Gevrey, Sur la nature analytique des solutions des ´ equations aux d´ eriv´ ees partielles, Ann

    M. Gevrey, Sur la nature analytique des solutions des ´ equations aux d´ eriv´ ees partielles, Ann. Sci. ´Ecole Norm. Sup. (3) 35 (1918), 129–190

    1918

  21. [27]

    Giacomin, Random Polymer Models (Imperial College Press, World Scientific, 2007)

    G. Giacomin, Random Polymer Models (Imperial College Press, World Scientific, 2007)

    2007

  22. [28]

    Giacomin, Disorder and critical phenomena through basic probability models , Lecture Notes in Mathematics 2025 (Springer, Heidelberg, 2011)

    G. Giacomin, Disorder and critical phenomena through basic probability models , Lecture Notes in Mathematics 2025 (Springer, Heidelberg, 2011)

    2025

  23. [30]

    Giacomin and M

    G. Giacomin and M. Khatib, Generalized Poland Scheraga denaturation model and two dimensional renewal processes, Stoch. Process. Their Appl. 127 (2017), 526–573

    2017

  24. [31]

    Giacomin and F.L

    G. Giacomin and F.L. Toninelli, The localized phase of disordered copolymers with adsorption, Lat. Am. J. Probab. 1 (2006), 149–180

    2006

  25. [32]

    Giacomin and F.L

    G. Giacomin and F.L. Toninelli, Smoothing effect of quenched disorder on polymer depinning transi- tions, Commun. Math. Phys. 266 (2006), 1–16

    2006

  26. [33]

    Giacomin and F.L

    G. Giacomin and F.L. Toninelli, Smoothing of depinning transitions for directed polymers with quenched disorder, Phys. Rev. Lett. 96 (2006), 070602

    2006

  27. [35]

    Godr` eche, Condensation for random variables conditioned by the value of their sum, J

    C. Godr` eche, Condensation for random variables conditioned by the value of their sum, J. Stat. Mech. Theory Exp. 6 (2019), 063207

    2019

  28. [36]

    Grosskinsky, P

    S. Grosskinsky, P. Chleboun, and G.M. Sch¨ utz, Instability of condensation in the zero-range process with random interaction, Phys. Rev. E 78 (2008), 030101

    2008

  29. [37]

    den Hollander, Large Deviations (American Mathematical Society, Providence, 2000)

    F. den Hollander, Large Deviations (American Mathematical Society, Providence, 2000)

    2000

  30. [38]

    den Hollander, Random polymers, Lecture Notes in Mathematics 1974 (Springer-Verlag, Berlin, 2009)

    F. den Hollander, Random polymers, Lecture Notes in Mathematics 1974 (Springer-Verlag, Berlin, 2009)

    1974

  31. [39]

    den Hollander and M

    F. den Hollander and M. Zamparo, Quenched large deviations in renewal theory, Stoch. Process. Their Appl. 175 (2024), 104414

    2024

  32. [40]

    Krantz and H.R

    S.G. Krantz and H.R. Parks, A primer of real analytic functions , 2nd edn (Birkh¨ user Advanced Texts, Boston, 2002)

    2002

  33. [41]

    Legrand, Influence of disorder on DNA denaturation: the disordered generalized Poland-Scheraga model, Electron

    A. Legrand, Influence of disorder on DNA denaturation: the disordered generalized Poland-Scheraga model, Electron. J. Probab. 26 (2021), 1–43

    2021

  34. [42]

    Mikosch and A

    T. Mikosch and A. V. Nagaev, Large deviations of heavy-tailed sums with applications in insurance, Extremes 1 (1998), 81–110

    1998

  35. [43]

    del Molino, P

    L.C.G. del Molino, P. Chleboun, and S. Grosskinsky, Condensation in randomly perturbed zero-range processes, J. Phys. A 45 (2012), 205001

    2012

  36. [44]

    Poland and H.A

    D. Poland and H.A. Scheraga, Theory of helix-coil transitions in biopolymers: statistical mechanical theory of order-disorder transitions in biological macromolecules (Academic Press, New York, 1970)

    1970

  37. [45]

    Wouts, Surface tension in the dilute Ising model

    M. Wouts, Surface tension in the dilute Ising model. The Wulff construction, Commun. Math. Phys. 289 (2009), 157–204

    2009

  38. [46]

    Velenik, Localization and delocalization of random interfaces, Probab

    Y. Velenik, Localization and delocalization of random interfaces, Probab. Surv. 3 (2006), 112–169

    2006

  39. [47]

    Vezzani, E

    A. Vezzani, E. Barkai and R. Burioni, Single-big-jump principle in physical modeling, Phys. Rev. E 100 (2019), 012108

    2019

  40. [48]

    Zamparo, Large deviations in renewal models of statistical mechanics, J

    M. Zamparo, Large deviations in renewal models of statistical mechanics, J. Phys. A: Math. Theor. 52 (2019), 495004

    2019

  41. [49]

    Zamparo, Critical fluctuations in renewal models of statistical mechanics, J

    M. Zamparo, Critical fluctuations in renewal models of statistical mechanics, J. Math. Phys.62 (2021), 113301

    2021

  42. [50]

    Zamparo, Large fluctuations and transport properties of the L´ evy–Lorentz gas, Ann

    M. Zamparo, Large fluctuations and transport properties of the L´ evy–Lorentz gas, Ann. Inst. H. Poincar´ e Probab. Statist.59 (2023), 621–661

    2023

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