# Mapping current and activity fluctuations in exclusion processes: consequences and open questions

### Submission summary

 As Contributors: Vivien Lecomte · Matthieu Vanicat Preprint link: scipost_202011_00003v1 Date submitted: 2020-11-04 11:03 Submitted by: Lecomte, Vivien Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

Considering the large deviations of activity and current in the Asymmetric Simple Exclusion Process (ASEP), we show that there exists a non-trivial correspondence between the joint scaled cumulant generating functions of activity and current of two ASEPs with different parameters. This mapping is obtained by applying a similarity transform on the deformed Markov matrix of the source model in order to obtain the deformed Markov matrix of the target model. We first derive this correspondence for periodic boundary conditions, and show in the diffusive scaling limit (corresponding to the Weakly Asymmetric Simple Exclusion Processes, or WASEP) how the mapping is expressed in the language of Macroscopic Fluctuation Theory (MFT). As an interesting specific case, we map the large deviations of current in the ASEP to the large deviations of activity in the SSEP, thereby uncovering a regime of Kardar–Parisi–Zhang in the distribution of activity in the SSEP. At large activity, particle configurations exhibit hyperuniformity [Jack et al., PRL 114 060601 (2015)]. Using results from quantum spin chain theory, we characterize the hyperuniform regime by evaluating the small wavenumber asymptotic behavior of the structure factor at half-filling. Conversely, we formulate from the MFT results a conjecture for a correlation function in spin chains at any fixed total magnetization (in the thermodynamic limit). In addition, we generalize the mapping to the case of two open ASEPs with boundary reservoirs, and we apply it in the WASEP limit in the MFT formalism. This mapping also allows us to find a symmetry-breaking dynamical phase transition (DPT) in the WASEP conditioned by activity, from the prior knowledge of a DPT in the WASEP conditioned by the current.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission scipost_202011_00003v2 on 21 January 2021

Submission scipost_202011_00003v1 on 4 November 2020

## Reports on this Submission

### Anonymous Report 2 on 2021-1-20 Invited Report

• Cite as: Anonymous, Report on arXiv:scipost_202011_00003v1, delivered 2021-01-20, doi: 10.21468/SciPost.Report.2435

### Strengths

1. Expands and completes our understanding of dynamical large deviations in simple exclusion processes (a fundamental basic model of non-equilibrium), specifically for fluctuations of the dynamical activity and current in the large deviation regime.
2. It does so via a simple (and very useful) relation between the tilted generators (for activity and current) of different SEPs, both for the periodic boundaries and for boundary driven cases.
3. It aggregates a decade or more of work in this problem. The paper is both conceptually clear and of high technical standard, combining exact methods for spin chains with MFT in the weakly asymmetric SEP.
4. Reveals a KPZ regime for activity fluctuations in the symmetric SEP.
5. Relates to the growing list of mappings in trajectory ensembles of non-equilibrium systems, such as Doob and gauge transformations, or mappings between equilibrium and non-equilibrium.

### Report

This paper deals with an important problem in non-equilibrium statistical mechanics, the precise characterization of fluctuations in the dynamics. The natural framework for this question is the method of large deviations, and simple exclusion processes (SEPs) are fundamental models. Perhaps "completes" is too strong a word as some questions remain, but the paper goes a long way to fully complement and clarify over a decade of work on the LDs in SEPs, work that has ranged from exact methods for spin chains, in the diffusive regime via MFT, and numerical simulations. How this fits all together is given in the summary of known and new results in Sec.3.1. This is an excellent piece of work of high standard. I recommend publication as is.

• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent

### Anonymous Report 1 on 2021-1-3 Invited Report

• Cite as: Anonymous, Report on arXiv:scipost_202011_00003v1, delivered 2021-01-02, doi: 10.21468/SciPost.Report.2352

### Strengths

1) Important new mapping for the ASEP

2) New formulas for LDF of activity

3) Relation with MFT and Spin Chains

4) A difficult open problem is raised in the diffusive regime. Can be viewed as a challenge in the field.

### Report

This manuscript is a very good piece of work. The authors have discovered
a very simple mapping that relates two deformed generators of the ASEP
with different parameters (asymmetries, fugacities of activity and current),
both in the periodic and the (finite) open case. As far as I can tell,
this elementary observation was unknown (which is almost embarrassing!):
it allows the authors to draw
many new relations and a lot of relevant information about exclusion
processes.
Many known results, scattered in the
literature of the last dozen years, can be restated within this
new perspective and appear now in a
coherent framework. In particular, I have found the section 3-7
(that relates the
structure factor in the hyperuniform phase with
ground state spin-spin correlations) particularly appealing.
Finally, this work, besides its original results, also
raises and states precise and important questions, that set interesting
challenges for the specialists.

\vskip 0.3cm

Here are some minor remarks on the manuscript:

\vskip 0.1cm

1) Section 3-1 describes various scaling regimes. In could be useful
if the authors could draw a table to summarize these results.
\vskip 0.1cm

\vskip 0.1cm
2) The expansion of ${\mathcal K}(s)$ mentioned after (3.20) is not clear.

\vskip 0.1cm
3) Can the KPZ scaling mentioned after (3.23) be related to a property of the
spectrum of SSEP ?

\vskip 0.1cm
4) What do the authors mean when they say after eq. (3-30) that they
checked numerically the existence of the divergence by diagonalizing the
evolution operator.

\vskip 0.1cm
5) A reference for eqs (4.18) could be useful.

\vskip 0.1cm
\vskip 0.1cm
I recommend publication of this manuscript in SCiPost without hesitation.

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: good
• grammar: good