Pith. sign in

REVIEW 2 major objections 4 minor 43 references

Applications of optimal transport to Dyson Brownian Motions and beyond

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A new optimal-transport comparison proves that β-Dyson Brownian motions with β≥2 and their scaling limits have exactly the sharp modulus of continuity of Brownian motion, uniformly across all layers, for a broad class of log-concave…

desk verdict A genuinely new and correct method: Caffarelli–Hargé comparison for log-concave perturbations yields Brownian-sharp, layer-uniform modulus estimates for DBM, Airy_β, KPZ, and OY; minor gaps only. read the letter →

arxiv 2412.17389 v4 pith:DQNLIKFC submitted 2024-12-23 math.PR

Xuan Wu This is my paper · ORCID
classification math.PR MSC 60B2060H1060G1749Q22
keywords DysonBrownianmotionlog-concaveperturbationsoptimaltransportCaffarellicontractionmodulusofcontinuityAirylineensembleKPZO'Connell-Yor
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

The paper claims that β-Dyson Brownian motions with β≥2, along with a broad class of random curve collections that are log-concave perturbations of Brownian motion, have exactly the same sharp modulus of continuity as a standard Brownian motion: their centered increments satisfy $\mathbb{E}|\hat L_j(t)-\hat L_j(s)|^p \le N_p |t-s|^{p/2}$ for every $p\ge1$, and the supremum over an interval has a Gaussian tail with constants independent of the layer. The proof is a direct comparison with Brownian motion, based on viewing the perturbed process as a reweighting of Wiener measure by $e^{-H}$ with a convex Hamiltonian $H$, and then applying Caffarelli's contraction theorem from optimal transport. The class $LC$ includes the β-Dyson Brownian motion for β≥2, the Airy$_β$ line ensemble, the KPZ line ensemble, and the O'Connell-Yor line ensemble, so the same estimates hold for all of them uniformly across every layer. If the argument is right, these are the sharp, layer-uniform continuity estimates that earlier work lacked, and they fill a gap in the program of constructing the directed landscape and the KPZ equation from line ensembles.

What carries the argument

The machinery is the class $LC$ of log-concave perturbations defined by convex Hamiltonians $H(z)=\sum_i f_i(z(t_i))+\int_a^b F(t,z(t))\,dt$ on Wiener space, and the key identity is Lemma 2.11: for any convex $H$, the partition function $Z_H(x,y)=\mathbb{E}_{x,y}[e^{-H(z)}]$ of a tilted Brownian bridge is log-concave in the endpoint data $(x,y)$. This log-concavity, transferred to finite-dimensional marginal densities via the Prékopa–Leindler theorem, makes the Radon–Nikodym derivative $d\hat\mu/d\hat\gamma$ log-concave (Lemma 2.4). Then Caffarelli's contraction theorem (the optimal-transport map pushing a Gaussian onto a log-concave perturbation is 1-Lipschitz), together with Hargé's convex/log-concave correlation inequality, gives the comparison $\mathbb{E}_\mu[g(w-\bar w_\mu)] \le \mathbb{E}_\gamma[g(w-\bar w_\gamma)]$ for every convex $g$ depending on finitely many times, which is exactly what yields the Brownian-style moment and tail bounds.

What would settle it

Compute, for a concrete convex Hamiltonian $H$, the partition function $Z_H(x,y)$ of a Brownian bridge on a fixed interval and check numerically whether $\log Z_H$ is concave in $(x,y)$; a single counterexample would falsify Lemma 2.11 and the whole theorem chain. Alternatively, simulate a $\beta=2$ Dyson Brownian motion with many particles and measure the variance of a centered single-particle increment over a small time step; if it exceeds $|t-s|$ by a non-negligible margin, Theorem 1.1(i) would be false.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 2.8: for every random continuous function $L$ in the class $LC$, the centered increment $\hat L_j(t)-\hat L_j(s)$ has $p$-th moment at most $N_p |t-s|^{p/2}$ (the Brownian value), and the normalized supremum over $[a,b]$ satisfies $\mathbb{P}\left(\sup_{t,s\in[a,b]}\frac{|\hat L_j(t)-\hat L_j(s)|}{\sqrt{|t-s|\log(2(b-a)/|t-s|)}}>K\right) \le C_1 e^{-C_2 K^2}$ with universal constants $C_1,C_2$ independent of the layer $j$. In particular, β-Dyson Brownian motions with $\beta\ge2$ satisfy the same sharp bounds as Brownian motion, uniformly in $j,t,s$, and the same holds after edge scaling for the Airy$_β$ line ensemble, for the KPZ line ensemble, and for the O'Connell-Yor line ensemble. The comparison is made by proving that finite-dimensional marginals of any $LC$ element are log-concave perturbations of Gaussian marginals, which brings the contraction theorem into play.

Load-bearing premise

The load-bearing premise is that the partition function of a Brownian bridge tilted by any convex Hamiltonian is log-concave in the bridge endpoints; if that failed for some convex Hamiltonian, the finite-dimensional densities would not be log-concave and the contraction comparison would break.

Editorial extensions

If this is right

  • For $\beta\ge2$, the $\beta$-Dyson Brownian motion and its edge-scaling limit Airy$_β$ satisfy uniform Gaussian-tail bounds on the modulus of continuity that are independent of the layer, with constants that do not grow in $j$.
  • The KPZ line ensemble, with $\hat X^T_j(t)=X^T_j(t)+2^{-1}t^2$, satisfies $\mathbb{E}\|\hat X^T_j\|^p_{\alpha,[a,b]} \le C(\alpha,p)(b-a)^{p/2-\alpha p}$, confirming Conjecture 1.12 in [Wu23b].
  • The finite-dimensional marginal distributions of the $\beta$-Dyson Brownian motion, the Airy$_β$ line ensemble, both with $\beta\ge2$, the O'Connell-Yor line ensemble, and the KPZ line ensemble are log-concave.
  • Because $LC$ is closed under distributional convergence, the estimates pass automatically to any distributional limit of log-concave perturbations, yielding an alternative short proof of tightness for the Airy$_β$ scaling limits.

Reading between the lines

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

  • The same optimal-transport comparison should apply to other log-concave random-matrix-type diffusions such as $\beta$-Laguerre and $\beta$-Jacobi processes, once their Girsanov Hamiltonians are shown convex; the paper notes these examples but leaves them open.
  • The layer-uniform Brownian-quality bounds for $\beta\ge2$ suggest that Airy$_β$ line ensembles are as regular as Brownian motion at small scales; this regularity may be the missing ingredient for constructing directed-landscape analogues for general $\beta$ from these ensembles.
  • One concrete test would be numerical: for a non-quadratic convex Hamiltonian, compute $\log Z_H(x,y)$ and check its Hessian in $(x,y)$; a negative eigenvalue would locate the failure of Lemma 2.11 rather than merely weakening the estimates.
  • Log-concavity of finite-dimensional marginals implies positive-correlation (FKG-type) properties for these line ensembles, which may lead to new monotonicity results not stated in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a new method, based on Caffarelli's contraction theorem and Hargé's convex/log-concave correlation inequality, to prove sharp and uniform modulus-of-continuity estimates for a class LC of random curve collections that are log-concave perturbations of Brownian motions or Brownian bridges, together with their distributional limits. The main result, Theorem 2.8, shows that for every member of LC, the centered increments satisfy Brownian-motion-type moment bounds, Gaussian tail bounds for a suitably normalized sup-norm, and Garsia-Rodemich-Rumsey type Hölder estimates, with constants uniform in the layer. The paper then proves that β-Dyson Brownian motions for β ≥ 2 belong to LC, and, by taking distributional limits, that the Airy_β line ensemble, the O'Connell-Yor line ensemble, and the KPZ line ensemble also belong to LC. Applications include new modulus-of-continuity estimates for the KPZ line ensemble and a short proof of tightness for the scaled Dyson Brownian motions.

Significance. If the results are fully established, this is a substantial contribution to the study of line ensembles and random matrix diffusions. The paper provides a unified framework that yields sharp, layer-uniform estimates matching those of Brownian motion for several important models, extending earlier work that was either restricted to β = 2 or had layer-dependent constants. The method is novel and conceptually clean, reducing the infinite-dimensional comparison to a finite-dimensional log-concavity statement (Lemma 2.11) that is checked via Prékopa–Leindler. A particular strength is that the proofs use no fitted parameters: all constants are explicit or universal, and the main estimates are derived from external theorems (Caffarelli, Hargé) rather than from self-cited results. The applications to the KPZ line ensemble and the O'Connell-Yor line ensemble are new and directly address an open gap in the construction of the fundamental solution to the KPZ equation.

major comments (2)
  1. [§3.1, proof of Corollary 1.3] The claim that √(β/2) A^β(t) belongs to LC is not justified. Proposition 3.1 establishes that the β-Dyson Brownian motion X^{β,N} itself is in LC, but the edge scaling (1.2) followed by normalization √(β/2) is an affine change of both space and time. The class LC is defined with respect to unit-diffusion Wiener measures and Hamiltonians of the form (2.1); it is not immediate that the pushforward of a member of LC under this transformation remains in LC. Since Corollaries 1.2 and 1.3 depend on applying Theorem 2.8 to the scaled process, the proof should either verify that the scaled base measure is a unit-diffusion Wiener measure and the transformed Hamiltonian remains of the form (2.1), or derive the estimate by direct scaling from Theorem 2.8 for X^{β,N}.
  2. [§2, Lemmas 2.2, 2.4, 2.12] Several statements that are used in the main proofs are omitted: the bridge counterpart of Lemma 2.2 (after (2.3)), the bridge counterpart of Lemma 2.4, and the entire Lemma 2.12. The bridge counterpart of Lemma 2.2 is used in the proof of Lemma 2.11 to reduce to non-negative Hamiltonians; Lemma 2.12 is used in the proof of Lemma 3.5 to show that jλ is log-concave; and the bridge counterpart of Lemma 2.4 is needed for Proposition 2.9(2.5) and hence for the bridge part of Theorem 2.8. These results are plausible variants of the proved arguments, but since they are load-bearing, the authors should supply the proofs or at least a detailed indication of the modifications required.
minor comments (4)
  1. [§3.2, Lemma 3.5] The displayed definition of H(z) appears to miss a minus logarithm; as written, H(z) = jλ(z(t0)) exp(−∫ F), which is not a Hamiltonian of the form (2.1). The intended H is presumably −log jλ(z(t0)) + ∫ F(z(s)) ds, consistent with the subsequent discussion of log-concavity of jλ.
  2. [§2, proof of Theorem 2.8(ii)] The symbol a is used both as a real parameter in [0,1/2) and as the left endpoint of the interval [a,b]; consider renaming the parameter (e.g., γ) to avoid confusion.
  3. [§3.2, Lemma 3.5] The sentence 'It is straightforward to check that for M (t) is an non-negative P0-martingale' contains a grammatical error and a stray period; more importantly, the martingale property is a nontrivial step in the Girsanov argument and should be justified by a reference or a short argument.
  4. [§3.1, proof of Proposition 3.1] The convergence e^{−Hε(z)} ↓ hβ(x∗) dµ/dγ(z) for γ-a.e. z is stated as straightforward, but a short explanation of why the approximating convex functions yield monotone convergence and why the limit vanishes on paths that leave the Weyl chamber would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Caffarelli–Hargé comparison is derived from external theorems; LC membership is proven by explicit Radon–Nikodym derivatives, and self-citations are not load-bearing.

full rationale

The derivation chain contains no step in which a claimed prediction is equivalent by construction or by definition to an input. The class LC is defined structurally as log-concave reweightings of Brownian motions or Brownian bridges (Definitions 2.3 and 2.6), not by the modulus-of-continuity bounds being proved. Membership of the β-Dyson Brownian motion in LC is established directly by an explicit Girsanov density (Lemma 3.2, Proposition 3.1), with the convexity of the relevant Hamiltonian checked in the paper. The central comparison, Theorem 2.8, is derived from Proposition 2.9, whose proof applies Caffarelli's contraction theorem and Hargé's inequality to finite-dimensional Gaussian marginals; the required log-concavity is supplied by Lemmas 2.4 and 2.11, proved via Lemma 2.10 and Prékopa–Leindler. None of these ingredients is a fitted quantity disguised as a prediction, and the universal constants are explicit. The self-citations are [Wu22], [Wu23a], and [Wu23b]; [Wu22] appears only as one reference in a list, [Wu23a] identifies an example not used in the main proof, and [Wu23b] contains a conjecture that Corollary 1.4 confirms rather than assumes. The distributional-limit inputs for the Airyβ and KPZ line ensembles are cited from independent sources ([Lan20, HZ24, GXZ24] and [CH16, Nic21]). No imported uniqueness theorem is invoked to force a choice, and no ansatz is smuggled in through the author's own prior work. Accordingly, the derivation is self-contained in the senses that matter for circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No constants are fitted to data; the universal constants C1, C2, C(alpha,p) are abstract. The derivation relies on standard optimal transport and convex geometry theorems, plus the cited existence and convergence results for the specific line ensembles.

assumptions (4)
  • standard math Caffarelli's contraction theorem and Hargé's convex-log-concave correlation inequality
    Used in Proposition 2.9 to compare expectations of convex finite-dimensional functionals under a Gaussian and under a probability measure with log-concave density with respect to it.
  • standard math Prékopa-Leindler theorem
    Used in Lemma 2.11, Lemma 2.12, and Corollary 2.7 to preserve log-concavity under integration and weak limits.
  • domain assumption Existence, uniqueness, and non-collision of the beta-Dyson Brownian motion and of the O'Connell-Yor diffusion, plus distributional convergence of edge scalings to Airy_beta and KPZ line ensembles
    Invoked in Propositions 3.1 and 3.4 and Corollaries 1.2 to 1.4; cited to AGZ10, O'C12, Lan20, HZ24, GXZ24, CH16, and Nic21.
  • domain assumption The Whittaker function representation j_lambda from BO11 and its use as an eigenfunction for the quantum Toda Hamiltonian
    Used in Lemma 3.5 to write the O'Connell-Yor process as a log-concave perturbation; the log-concavity of j_lambda is then derived by applying Lemma 2.12 and taking the infinite time limit.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Applications of optimal transport to Dyson Brownian Motions and beyond." pith.science (2026). https://pith.science/paper/DQNLIKFC

@misc{pith2026241217389,
  author       = {Pith},
  title        = {Pith review of: Applications of optimal transport to Dyson Brownian Motions and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQNLIKFC}},
  note         = {Machine review of arXiv:2412.17389}
}
abstract

We develop a new method based on Caffarelli's contraction theorem in optimal transport to obtain sharp and uniform modulus of continuity estimates for $\beta$-Dyson Brownian motions with $\beta \geq 2$. Our method extends to a large class of random curve collections, which can be viewed as log-concave perturbations of Brownian motions, including the $\beta$-Dyson Brownian motion, the Air$\text{y}_{\beta}$ line ensemble, the KPZ line ensemble, and the O'Connell-Yor line ensemble.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 31 canonical work pages

  1. [1]

    Ferrari, and Pierre van Moerbeke

    Mark Adler, Patrik L. Ferrari, and Pierre van Moerbeke. Airy processes with wanderers and new universality classes. Ann. Probab. , 38(2):714--769, 2010

    work page 2010

  2. [2]

    Anderson, Alice Guionnet, and Ofer Zeitouni

    Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices , volume 118 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2010

    2010

  3. [3]

    Strong characterization for the A iry line ensemble

    Amol Aggarwal and Jiaoyang Huang. Strong characterization for the A iry line ensemble. 2023. arXiv:2308.11908

    arXiv 2023

  4. [4]

    Billingsley

    P. Billingsley. Convergence of probability measures . Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication

    1999

  5. [5]

    Log-concavity in one-dimensional coulomb gases and related ensembles

    Jnaneshwar Baslingker, Manjunath Krishnapur, and Mokshay Madiman. Log-concavity in one-dimensional coulomb gases and related ensembles. 2024. arXiv:2412.15116

    arXiv 2024

  6. [6]

    Exponential functionals of B rownian motion and class-one W hittaker functions

    Fabrice Baudoin and Neil O'Connell. Exponential functionals of B rownian motion and class-one W hittaker functions. Ann. Inst. Henri Poincar\'e Probab. Stat. , 47(4):1096--1120, 2011

    work page 2011

  7. [7]

    Caffarelli

    Luis A. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys. , 214(3):547--563, 2000

    work page 2000

  8. [8]

    Stochastic L oewner evolution and D yson's circular ensembles

    John Cardy. Stochastic L oewner evolution and D yson's circular ensembles. J. Phys. A , 36(24):L379--L386, 2003

    work page 2003

Show all 43 references
  1. [9]

    Brownian G ibbs property for A iry line ensembles

    Ivan Corwin and Alan Hammond. Brownian G ibbs property for A iry line ensembles. Invent. Math. , 195(2):441--508, 2014

    2014

  2. [10]

    K PZ line ensemble

    Ivan Corwin and Alan Hammond. K PZ line ensemble. Probab. Theory Related Fields , 166(1-2):67--185, 2016

    2016

  3. [11]

    Airy wanderer line ensembles

    Evgeni Dimitrov. Airy wanderer line ensembles. 2024. arXiv:2408.08445

    2024 arXiv

  4. [12]

    Unimodality, convexity, and applications

    Sudhakar Dharmadhikari and Kumar Joag-Dev. Unimodality, convexity, and applications . Probability and Mathematical Statistics. Academic Press, Inc., Boston, MA, 1988

    1988

  5. [13]

    Characterization of B rownian G ibbsian line ensembles

    Evgeni Dimitrov and Konstantin Matetski. Characterization of B rownian G ibbsian line ensembles. Ann. Probab. , 49(5):2477--2529, 2021

    2021

  6. [14]

    Uniform convergence to the airy line ensemble

    Duncan Dauvergne, Mihai Nica, and B \'a lint Vir \'a g. Uniform convergence to the airy line ensemble. arXiv preprint , 2019

    2019

  7. [15]

    The directed landscape

    Duncan Dauvergne, Janosch Ortmann, and Bálint Virág. The directed landscape. Acta Mathematica , 229(2):201–285, 2022

    2022

  8. [16]

    Bulk properties of the A iry line ensemble

    Duncan Dauvergne and B\'alint Vir\'ag. Bulk properties of the A iry line ensemble. Ann. Probab. , 49(4):1738--1777, 2021

    2021

  9. [17]

    The scaling limit of the longest increasing subsequence

    Duncan Dauvergne and Bálint Virág. The scaling limit of the longest increasing subsequence. 2022. arXiv:2104.08210

    2022 arXiv

  10. [18]

    Freeman J. Dyson. A B rownian-motion model for the eigenvalues of a random matrix. J. Mathematical Phys. , 3:1191--1198, 1962

    1962

  11. [19]

    Feyel and A

    D. Feyel and A. S. \"Ust\"unel. Monge- K antorovitch measure transportation and M onge- A mp\`ere equation on W iener space. Probab. Theory Related Fields , 128(3):347--385, 2004

    2004

  12. [20]

    A. M. Garsia, E. Rodemich, and H. Rumsey, Jr. A real variable lemma and the continuity of paths of some G aussian processes. Indiana Univ. Math. J. , 20:565--578, 1970/71

    1970

  13. [21]

    Airy _ line ensemble and its L aplace transform

    Vadim Gorin, Jiaming Xu, and Lingfu Zhang. Airy _ line ensemble and its L aplace transform. 2024. arXiv:2411.10829

    2024 arXiv

  14. [22]

    A convex/log-concave correlation inequality for G aussian measure and an application to abstract W iener spaces

    Gilles Harg\'e. A convex/log-concave correlation inequality for G aussian measure and an application to abstract W iener spaces. Probab. Theory Related Fields , 130(3):415--440, 2004

    2004

  15. [23]

    A convergence framework for A iry _ line ensemble via pole evolution

    Jiaoyang Huang and Lingfu Zhang. A convergence framework for A iry _ line ensemble via pole evolution. 2024. arXiv:2411.10586

    2024 arXiv

  16. [24]

    Gaussian H ilbert spaces , volume 129 of Cambridge Tracts in Mathematics

    Svante Janson. Gaussian H ilbert spaces , volume 129 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1997

    1997

  17. [25]

    Dynamic scaling of growing interfaces

    Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. , 56:889--892, 3 1986

    1986

  18. [26]

    Edge scaling limit of D yson B rownian motion at equilibrium for general 1

    Benjamin Landon. Edge scaling limit of D yson B rownian motion at equilibrium for general 1 . 2020. arXiv:2009.11176

    2020 arXiv

  19. [27]

    The B rownian transport map

    Dan Mikulincer and Yair Shenfeld. The B rownian transport map. Probab. Theory Related Fields , 190(1-2):379--444, 2024

    2024

  20. [28]

    Intermediate disorder limits for multi-layer semi-discrete directed polymers

    Mihai Nica. Intermediate disorder limits for multi-layer semi-discrete directed polymers. Electron. J. Probab. , 26:Paper No. 62, 50, 2021

    2021

  21. [29]

    Directed polymers and the quantum T oda lattice

    Neil O'Connell. Directed polymers and the quantum T oda lattice. Ann. Probab. , 40(2):437--458, 2012

    2012

  22. [30]

    A multi-layer extension of the stochastic heat equation

    Neil O'Connell and Jon Warren. A multi-layer extension of the stochastic heat equation. Comm. Math. Phys. , 341(1):1--33, 2016

    2016

  23. [31]

    Brownian analogues of B urke's theorem

    Neil O'Connell and Marc Yor. Brownian analogues of B urke's theorem. Stochastic Process. Appl. , 96(2):285--304, 2001

    2001

  24. [32]

    Scale invariance of the PNG droplet and the A iry process

    Michael Pr\"ahofer and Herbert Spohn. Scale invariance of the PNG droplet and the A iry process. volume 108, pages 1071--1106. 2002. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays

    2002

  25. [33]

    J. D. Quastel. The K ardar- P arisi- Z hang equation and universality class. In X VII th I nternational C ongress on M athematical P hysics , pages 113--133. World Sci. Publ., Hackensack, NJ, 2014

    2014

  26. [34]

    Ram\'irez, Brian Rider, and B\'alint Vir\'ag

    Jos\'e A. Ram\'irez, Brian Rider, and B\'alint Vir\'ag. Beta ensembles, stochastic A iry spectrum, and a diffusion. J. Amer. Math. Soc. , 24(4):919--944, 2011

    2011

  27. [35]

    L. C. G. Rogers and Z. Shi. Interacting B rownian particles and the W igner law. Probab. Theory Related Fields , 95(4):555--570, 1993

    1993

  28. [36]

    Continuous martingales and B rownian motion , volume 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Daniel Revuz and Marc Yor. Continuous martingales and B rownian motion , volume 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, third edition, 1999

    1999

  29. [37]

    Convexity , volume 187 of Cambridge Tracts in Mathematics

    Barry Simon. Convexity , volume 187 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2011. An analytic viewpoint

    2011

  30. [38]

    Stroock and S

    Daniel W. Stroock and S. R. Srinivasa Varadhan. Multidimensional diffusion processes . Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1997 edition

    2006

  31. [39]

    Infinite dimensional stochastic differential equations for D yson's model

    Li-Cheng Tsai. Infinite dimensional stochastic differential equations for D yson's model. Probab. Theory Related Fields , 166(3-4):801--850, 2016

    2016

  32. [40]

    Tracy and Harold Widom

    Craig A. Tracy and Harold Widom. Level-spacing distributions and the A iry kernel. Comm. Math. Phys. , 159(1):151--174, 1994

    1994

  33. [41]

    Convergence of the KPZ line ensemble

    Xuan Wu. Convergence of the KPZ line ensemble. International Mathematics Research Notices , 2022

    2022

  34. [42]

    The B essel line ensemble

    Xuan Wu. The B essel line ensemble. Electron. J. Probab. , 28:Paper No. 77, 50, 2023

    2023

  35. [43]

    The KPZ equation and the directed landscape

    Xuan Wu. The KPZ equation and the directed landscape. 2023. arXiv:2301.00547

    2023 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.