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Differential Equations for the KPZ and Periodic KPZ Fixed Points

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Abstract

The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for the height function of models in the KPZ universality class. Similarly, the periodic KPZ fixed point is a conjectured universal field for spatially periodic models. For both fields, their multi-point distributions in the space-time domain have been computed recently. We show that for the case of the narrow-wedge initial condition, these multi-point distributions can be expressed in terms of so-called integrable operators. We then consider a class of operators that include the ones arising from the KPZ and the periodic KPZ fixed points, and find that they are related to various matrix integrable differential equations such as coupled matrix mKdV equations, coupled matrix NLS equations with complex time, and matrix KP-II equations. When applied to the KPZ fixed points, our results extend previously known differential equations for one-point distributions and equal-time, multi-position distributions to multi-time, multi-position setup.

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Notes

  1. The Airy\(_2\) process is often regarded as an evolution. The time in the Airy\(_2\) process corresponds to the position of the KPZ fixed point.

  2. The condition \(\textsf {H}(u,u)=0\) is not strictly necessarily. We included it here to simplify technical details in the theory, especially when \(\Omega \) is a discrete set. This simplification is enough for our purposes regarding the (periodic) KPZ fixed points.

  3. The usual KP-II equation is \( \partial _x \left( \partial _t w + 6w \partial _x w + \partial _{xxx} w \right) + 3\partial _{yy} w=0\) for a scalar function w. When \(m=1\), the x-derivative of Equation (2.14) becomes this equation if we set \(w(t,y,x)=2u(-4t,y,x)\).

  4. The paper also obtained the result for general initial conditions.

  5. In the paper [8], the roles of \(\textsf {r}\) and \(\textsf {p}\) are switched.

  6. The published version of the system from [30] has a typo which was later fixed in the ArXiv version (arXiv:ArXiv:0302033v4, see Equations (1)–(3) and the last two displayed equations in page 3 therein). Our discussion here uses the latest Arxiv version.

  7. The correspondence of terms between [26, Theorem 2.5] and here is \(\psi =\partial _{x}M\), \({\varvec{\Psi }}=M\), \(r=x\), and the variables \(y\) coincide.

  8. We can also consider similar to (2.10). Then the discussions below still hold if we define the parameters \(x^{(i)}_\ell \), \(i=1, \cdots , |\textsf {S}|\), \(\ell =1, 2,3\) appropriately. For the convenience of presentation, we state results only when here.

  9. To compute it, note, for example, that \(\partial ^3 u \partial ^{-2}= u \partial + 3 (\partial u) + 3(\partial ^2 u) \partial ^{-1} + (\partial ^3 u)\partial ^{-2}\) and \(\partial ^{-1} u \partial ^2=u\partial - (\partial u)+ (\partial ^2 u) \partial ^{-1}-\cdots \) from the product rule of derivatives, and thus, \((\partial ^3 u \partial ^{-1})_+= u \partial ^2+ 3 (\partial u) \partial \) and \((\partial ^{-1} u \partial ^2)+=u\partial - (\partial u)\) for any function u.

  10. This equation is the special case when \(\alpha =0\) in equation (113) of [20]. The full Sato equations involve pseudo-differential operators parametrized by \(\alpha \in \mathbb {Z}^{n-1}\). The multi-component KP hierarchy may also involve infinitely many parameters \(x^{(i)}_\ell \), \(i\in \{1, \cdots , n\}\), \(\ell \in \{1, 2, \cdots \}\). Here we have the special case when \(\ell \in \{1,2, 3\}\).

  11. In [23, Definition 2.25] the formulas \(Q_1(j)\) and \(Q_2(j)\) are used. Here, we use the function \(\textsf {Q}_{j}^\mathrm {(KPZ)}(z)=-Q_2(j) \chi _{\Omega _{1,j}}(z)+ Q_1(j) \chi _{\Omega _{2,j}}(z)\).

  12. The factor \(\textsf {P}_j^\mathrm {(KPZ)}\) is not present in the kernels in [23, Equation (5) and Definition 2.4] This is because the \(L^2\) space is weighted. Here we use \(L^2(\textrm{d}z)\) space instead and include \(\textsf {P}_j^\mathrm {(KPZ)}\) as a part of the kernel.

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Acknowledgements

The work of Baik was supported in part by NSF grant DMS-1954790. Silva acknowledges his current support by São Paulo Research Foundation under grants # 2019/16062-1 and # 2020/02506-2, and by Brazilian National Council for Scientific and Technological Development (CNPq) under grant # 315256/2020-6. Prokhorov was supported by NSF MSPRF grant DMS-2103354, NSF grant DMS-1928930, and RSF grant 22-11-00070. The part of the work was done while Prokhorov was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Jinho Baik & Andrei Prokhorov

  2. St. Petersburg State University, Universitetskaya emb. 7/9, St. Petersburg, Russia, 199034

    Andrei Prokhorov

  3. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, 13566-590, Brazil

    Guilherme L. F. Silva

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  1. Jinho Baik
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Correspondence to Guilherme L. F. Silva.

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Baik, J., Prokhorov, A. & Silva, G.L.F. Differential Equations for the KPZ and Periodic KPZ Fixed Points. Commun. Math. Phys. 401, 1753–1806 (2023). https://doi.org/10.1007/s00220-023-04683-z

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