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KPZ fluctuations in finite volume

by Sylvain Prolhac

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Submission summary

Authors (as registered SciPost users): Sylvain Prolhac
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-05-14
Date submitted: 2024-05-03 15:20
Submitted by: Prolhac, Sylvain
Submitted to: SciPost Physics Lecture Notes
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


These lecture notes, adapted from the habilitation thesis of the author, survey in a first part various exact results obtained in the past few decades about KPZ fluctuations in one dimension, with a special focus on finite volume effects describing the relaxation to its stationary state of a finite system starting from a given initial condition. The second part is more specifically devoted to an approach allowing to express in a simple way the statistics of the current in the totally asymmetric simple exclusion process in terms of a contour integral on a compact Riemann surface, whose infinite genus limit leads to KPZ fluctuations in finite volume.

List of changes

Changes in response to referee 1:
1 - the KPZ equation has been written already in the introduction.
2 - the fact that other estimators for the width of the interface do verify the Family-Vicsek scaling is now mentioned.
3 - the misprint "Cole-Hops" has been corrected.
4 - the sentence about anti-shocks was too convoluted. It has been rewritten.
5 - for TASEP with open boundaries, it should now be clearer why the low and high density phases must be excluded, on the time scale $t\sim L^{3/2}$.
6 - I agree with the referee that writing down explicit formulas here would be nice, but this would require to set up somewhat heavy notations. Since these formulas are not used in the following, are only tangentially related to my main point about KPZ fluctuations in finite volume, and can be found in the references, I would rather leave them out.
7 - an indication for the ordering of times has been added in the caption of the figure.

Changes in response to referee 2:
1 - the sentence was not so clear, I have rewritten.
2 - misprint corrected.
3 - formulation improved.
4 - none, the height increments are correct as stated.
5 - misprint corrected.
6 - the formulation has been improved.
7 - the mention of the Riemann surface $R_{KPZ}$ has been removed from section 2.4.3, but kept and expanded in the discussion below (21). It is now mentioned there how this is connected to analytic continuation between the eigenstates.
8 - misprint corrected.
9 - misprint corrected.
10 - footnote 11 has been rewritten and expanded, and figure 12 has been added to explain visually the difference between singular and exceptional points (or more generally ramified points with respect to some parameter).

Other changes:
- page 9: reference [71], where finite volume effects are discussed for condensates out of equilibrium, has been added.
- page 17: reference [227] has been updated (the discrepancy with the replica result has been resolved in the newer version) and moved to section 2.4.1, as [372].
- reference [461] has been added.
- figures 18 and 19 have been corrected (problem with the LaTeX interpreter)

Published as SciPost Phys. Lect. Notes 81 (2024)

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