Parisi's formula is a Hamilton-Jacobi equation in Wasserstein space

Jean-Christophe Mourrat

arxiv: 1906.08471 · v2 · pith:OTRTU23Hnew · submitted 2019-06-20 · 🧮 math.PR · cond-mat.dis-nn

Parisi's formula is a Hamilton-Jacobi equation in Wasserstein space

Jean-Christophe Mourrat This is my paper

Pith reviewed 2026-05-25 19:36 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.dis-nn

keywords Parisi formulaHamilton-Jacobi equationWasserstein spacespin glassesfree energymean-field modelsprobability measures

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The pith

Parisi's formula can be recast as the solution of a Hamilton-Jacobi equation in Wasserstein space.

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

Parisi's formula gives a variational expression for the infinite-volume limit of the free energy in mean-field spin glass models. The paper shows that this same quantity satisfies a Hamilton-Jacobi equation whose state variable lives in the Wasserstein space of probability measures on the positive half-line. The identification treats the validity of Parisi's formula as given and works entirely within the Wasserstein geometry to obtain the PDE. A reader might care because the PDE formulation supplies an alternative way to characterize or compute the free energy limit using tools from optimal transport and Hamilton-Jacobi theory.

Core claim

Parisi's formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We show that this quantity can be recast as the solution of a Hamilton-Jacobi equation in the Wasserstein space of probability measures on the positive half-line.

What carries the argument

The Hamilton-Jacobi equation in the Wasserstein space of probability measures on the positive half-line, which the free energy functional is shown to satisfy.

If this is right

The Parisi formula satisfies the Hamilton-Jacobi equation in Wasserstein space.
The free energy limit admits a characterization as the unique solution to this PDE.
Standard PDE techniques in Wasserstein space become applicable to the spin-glass free energy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The PDE perspective may allow transferring regularity or uniqueness results from Hamilton-Jacobi theory back to statements about the spin-glass model.
Similar recastings could be attempted for other variational problems whose state space is a space of measures.
Numerical schemes that evolve measures under Wasserstein geometry might be used to approximate the value of Parisi's formula.

Load-bearing premise

The infinite-volume limit of the free energy exists and equals exactly the expression given by Parisi's formula.

What would settle it

An explicit check that the functional defined by Parisi's formula does not solve the stated Hamilton-Jacobi equation when the Wasserstein distance and the associated Hamiltonian are computed directly.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Mourrat shows that the known Parisi formula satisfies a Hamilton-Jacobi equation in Wasserstein space over measures on the half-line.

read the letter

The main point is that Mourrat takes the explicit Parisi formula for the infinite-volume free energy and shows it solves a first-order Hamilton-Jacobi equation when the state space is the Wasserstein space of probability measures on the positive half-line. This is the new piece: a direct identification between the variational expression and the PDE in that geometry. The paper works from the formula as given and derives the PDE property without extra fitting or self-referential steps. That connection is clean and uses the structure of the overlap measures in a natural way. It could let people import comparison principles or viscosity techniques from the PDE side to study the functional. The argument appears self-contained and builds on Talagrand's theorem as background rather than trying to reprove the limit from the PDE. The only minor soft spot is that the result is a characterization of the already-known functional rather than an independent existence proof, but the paper does not claim otherwise. No hidden assumptions about regularity or post-hoc choices show up in the setup. This is aimed at people who already work on mean-field spin glasses and want analytic tools, or at analysts comfortable with Wasserstein geometry who might want a concrete example. A reader outside those two circles will probably not get much. The thinking is clear and the claim is precise, so the paper deserves a serious referee even if the payoff is mostly for specialists.

Referee Report

0 major / 2 minor

Summary. The paper claims that Parisi's formula, which gives the infinite-volume limit of the free energy for mean-field spin glass models (taken as given via Talagrand's theorem), can be recast exactly as the solution of a Hamilton-Jacobi equation posed in the Wasserstein space of probability measures supported on the positive half-line.

Significance. If the equivalence is rigorously established, the result supplies a new functional-analytic perspective on the Parisi formula by embedding it in the geometry of Wasserstein space. This may allow viscosity-solution techniques or optimal-transport tools to be applied to questions in spin-glass theory without altering the underlying variational problem. The manuscript correctly treats the existence of the limit as background and focuses on the recasting step.

minor comments (2)

[Introduction] The precise statement of the Hamilton-Jacobi equation (including the Hamiltonian and the initial condition) should be displayed as a numbered display equation early in the introduction or in a dedicated preliminary section to make the central claim immediately verifiable.
Notation for the Wasserstein metric, the space P(R_+), and the test functions used in the viscosity solution definition should be collected in a short notation table or paragraph to assist readers outside optimal transport.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper takes the existence of the infinite-volume limit and its equality to Parisi's formula as background (Talagrand's theorem) and shows that this known functional satisfies a Hamilton-Jacobi equation in Wasserstein space. This is a mathematical identification of properties of the functional, not a derivation that reduces the result to fitted inputs, self-definitions, or load-bearing self-citations. The Wasserstein-space analysis is independent of the limit's existence proof and does not presuppose the target PDE solution by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the claim rests on the prior existence of Parisi's formula and the standard definition of Wasserstein space.

pith-pipeline@v0.9.0 · 5557 in / 1055 out tokens · 24698 ms · 2026-05-25T19:36:15.747837+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that this quantity can be recast as the solution of a Hamilton-Jacobi equation in the Wasserstein space of probability measures on the positive half-line... Hopf-Lax formula f(t, μ) := sup_ν (ψ(ν) − t E[ξ*(Xν − Xμ / t)])
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 (Hamilton-Jacobi representation of Parisi formula)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 3 internal anchors

[1]

Ambrosio and J

L. Ambrosio and J. Feng. On a class of ﬁrst order Hamilton-Jacobi equations in metric spaces. J. Diﬀerential Equations , 256(7):2194–2245, 2014

work page 2014
[2]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient ﬂows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, second edition, 2008

work page 2008
[3]

Barra, G

A. Barra, G. Del Ferraro, and D. Tantari. Mean ﬁeld spin glasses treated with PDE techniques. Eur. Phys. J. B , 86(7):Art. 332, 10, 2013

work page 2013
[4]

Barra, A

A. Barra, A. Di Biasio, and F. Guerra. Replica symmetry breaking in mean-ﬁeld spin glasses through the Hamilton-Jacobi technique. J. Stat. Mech. Theory Exp. , (9):P09006, 22, 2010

work page 2010
[5]

S. H. Benton, Jr. The Hamilton-Jacobi equation. Academic Press, New York-London, 1977

work page 1977
[6]

J. G. Brankov and V. A. Zagrebnov. On the description of the phase transition in the Husimi- Temperley model. J. Phys. A , 16(10):2217–2224, 1983

work page 1983
[7]

M. G. Crandall and P.-L. Lions. Hamilton-Jacobi equations in inﬁnite dimensions. I. Uniqueness of viscosity solutions. J. Funct. Anal., 62(3):379–396, 1985

work page 1985
[8]

L. C. Evans. Partial diﬀerential equations , volume 19 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2010

work page 2010
[9]

Gangbo, T

W. Gangbo, T. Nguyen, and A. Tudorascu. Hamilton-Jacobi equations in the Wasserstein space. Methods Appl. Anal., 15(2):155–183, 2008

work page 2008
[10]

F. Guerra. Sum rules for the free energy in the mean ﬁeld spin glass model. Fields Institute Communications, 30:161, 2001

work page 2001
[11]

F. Guerra. Broken replica symmetry bounds in the mean ﬁeld spin glass model. Comm. Math. Phys., 233(1):1–12, 2003

work page 2003
[12]

M´ ezard, G

M. M´ ezard, G. Parisi, and M. Virasoro.Spin glass theory and beyond: an introduction to the replica method and its applications , volume 9. World Scientiﬁc Publishing Company, 1987

work page 1987
[13]

J.-C. Mourrat. Hamilton-Jacobi equations for mean-ﬁeld disordered systems. Preprint, arXiv:1811.01432

work page internal anchor Pith review Pith/arXiv arXiv
[14]

J.-C. Mourrat. Hamilton-Jacobi equations for ﬁnite-rank matrix inference. Preprint, arXiv:1904.05294

work page internal anchor Pith review Pith/arXiv arXiv 1904
[15]

C. Newman. Percolation theory: A selective survey of rigorous results. In Advances in multiphase ﬂow and related problems . SIAM, 1986

work page 1986
[16]

Panchenko

D. Panchenko. The Sherrington-Kirkpatrick model . Springer Monographs in Mathematics. Springer, New York, 2013. PARISI’S FORMULA IS A HAMILTON-JACOBI EQUATION IN W ASSERSTEIN SPACE 19

work page 2013
[17]

Guerra's interpolation using Derrida-Ruelle cascades

D. Panchenko and M. Talagrand. Guerra’s interpolation using Derrida-Ruelle cascades. Unpub- lished manuscript, arXiv:0708.3641 (2007)

work page internal anchor Pith review Pith/arXiv arXiv 2007
[18]

G. Parisi. A sequence of approximated solutions to the SK model for spin glasses. J. Phys. A , 13(4):L115–L121, 1980

work page 1980
[19]

M. M. Rao and Z. D. Ren. Theory of Orlicz spaces , volume 146 of Monographs and Textbooks in Pure and Applied Mathematics . Marcel Dekker, Inc., New York, 1991

work page 1991
[20]

Talagrand

M. Talagrand. Free energy of the spherical mean ﬁeld model. Probab. Theory Related Fields, 134(3):339–382, 2006

work page 2006
[21]

Talagrand

M. Talagrand. The Parisi formula. Ann. of Math. (2) , 163(1):221–263, 2006

work page 2006
[22]

Talagrand

M. Talagrand. Mean ﬁeld models for spin glasses. Volume I , volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer-Verlag, Berlin, 2011

work page 2011
[23]

Talagrand

M. Talagrand. Mean ﬁeld models for spin glasses. Volume II , volume 55 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer, Heidelberg, 2011

work page 2011
[24]

W. P. Thurston. On proof and progress in mathematics.Bull. Amer. Math. Soc. (N.S.), 30(2):161– 177, 1994

work page 1994
[25]

C. Villani. Topics in optimal transportation , volume 58 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2003. (J.-C. Mourrat) DMA, Ecole normale sup´erieure, CNRS, PSL University, Paris, France E-mail address: mourrat@dma.ens.fr

work page 2003

[1] [1]

Ambrosio and J

L. Ambrosio and J. Feng. On a class of ﬁrst order Hamilton-Jacobi equations in metric spaces. J. Diﬀerential Equations , 256(7):2194–2245, 2014

work page 2014

[2] [2]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient ﬂows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, second edition, 2008

work page 2008

[3] [3]

Barra, G

A. Barra, G. Del Ferraro, and D. Tantari. Mean ﬁeld spin glasses treated with PDE techniques. Eur. Phys. J. B , 86(7):Art. 332, 10, 2013

work page 2013

[4] [4]

Barra, A

A. Barra, A. Di Biasio, and F. Guerra. Replica symmetry breaking in mean-ﬁeld spin glasses through the Hamilton-Jacobi technique. J. Stat. Mech. Theory Exp. , (9):P09006, 22, 2010

work page 2010

[5] [5]

S. H. Benton, Jr. The Hamilton-Jacobi equation. Academic Press, New York-London, 1977

work page 1977

[6] [6]

J. G. Brankov and V. A. Zagrebnov. On the description of the phase transition in the Husimi- Temperley model. J. Phys. A , 16(10):2217–2224, 1983

work page 1983

[7] [7]

M. G. Crandall and P.-L. Lions. Hamilton-Jacobi equations in inﬁnite dimensions. I. Uniqueness of viscosity solutions. J. Funct. Anal., 62(3):379–396, 1985

work page 1985

[8] [8]

L. C. Evans. Partial diﬀerential equations , volume 19 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2010

work page 2010

[9] [9]

Gangbo, T

W. Gangbo, T. Nguyen, and A. Tudorascu. Hamilton-Jacobi equations in the Wasserstein space. Methods Appl. Anal., 15(2):155–183, 2008

work page 2008

[10] [10]

F. Guerra. Sum rules for the free energy in the mean ﬁeld spin glass model. Fields Institute Communications, 30:161, 2001

work page 2001

[11] [11]

F. Guerra. Broken replica symmetry bounds in the mean ﬁeld spin glass model. Comm. Math. Phys., 233(1):1–12, 2003

work page 2003

[12] [12]

M´ ezard, G

M. M´ ezard, G. Parisi, and M. Virasoro.Spin glass theory and beyond: an introduction to the replica method and its applications , volume 9. World Scientiﬁc Publishing Company, 1987

work page 1987

[13] [13]

J.-C. Mourrat. Hamilton-Jacobi equations for mean-ﬁeld disordered systems. Preprint, arXiv:1811.01432

work page internal anchor Pith review Pith/arXiv arXiv

[14] [14]

J.-C. Mourrat. Hamilton-Jacobi equations for ﬁnite-rank matrix inference. Preprint, arXiv:1904.05294

work page internal anchor Pith review Pith/arXiv arXiv 1904

[15] [15]

C. Newman. Percolation theory: A selective survey of rigorous results. In Advances in multiphase ﬂow and related problems . SIAM, 1986

work page 1986

[16] [16]

Panchenko

D. Panchenko. The Sherrington-Kirkpatrick model . Springer Monographs in Mathematics. Springer, New York, 2013. PARISI’S FORMULA IS A HAMILTON-JACOBI EQUATION IN W ASSERSTEIN SPACE 19

work page 2013

[17] [17]

Guerra's interpolation using Derrida-Ruelle cascades

D. Panchenko and M. Talagrand. Guerra’s interpolation using Derrida-Ruelle cascades. Unpub- lished manuscript, arXiv:0708.3641 (2007)

work page internal anchor Pith review Pith/arXiv arXiv 2007

[18] [18]

G. Parisi. A sequence of approximated solutions to the SK model for spin glasses. J. Phys. A , 13(4):L115–L121, 1980

work page 1980

[19] [19]

M. M. Rao and Z. D. Ren. Theory of Orlicz spaces , volume 146 of Monographs and Textbooks in Pure and Applied Mathematics . Marcel Dekker, Inc., New York, 1991

work page 1991

[20] [20]

Talagrand

M. Talagrand. Free energy of the spherical mean ﬁeld model. Probab. Theory Related Fields, 134(3):339–382, 2006

work page 2006

[21] [21]

Talagrand

M. Talagrand. The Parisi formula. Ann. of Math. (2) , 163(1):221–263, 2006

work page 2006

[22] [22]

Talagrand

M. Talagrand. Mean ﬁeld models for spin glasses. Volume I , volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer-Verlag, Berlin, 2011

work page 2011

[23] [23]

Talagrand

M. Talagrand. Mean ﬁeld models for spin glasses. Volume II , volume 55 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer, Heidelberg, 2011

work page 2011

[24] [24]

W. P. Thurston. On proof and progress in mathematics.Bull. Amer. Math. Soc. (N.S.), 30(2):161– 177, 1994

work page 1994

[25] [25]

C. Villani. Topics in optimal transportation , volume 58 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2003. (J.-C. Mourrat) DMA, Ecole normale sup´erieure, CNRS, PSL University, Paris, France E-mail address: mourrat@dma.ens.fr

work page 2003