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

### Submission summary

 As Contributors: Vivien Lecomte · Matthieu Vanicat Preprint link: scipost_202011_00003v2 Date accepted: 2021-01-27 Date submitted: 2021-01-21 22:47 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.

Published as SciPost Phys. 10, 028 (2021)

We are grateful to the referee for their positive assessment of our manuscript.

Referee’s comment: 1) Section 3-1 describes various scaling regimes. In could be useful if the authors could draw a table to summarize these results.

Our answer: We have improved the presentation of the list so that it clearly distinguishes the different regimes in s. We believe it will play the role of the table suggested by the referee, by gathering the definitions of the regimes of $s$ and the behaviour of the SCFG in a summarized manner.

Referee’s comment: 2) The expansion of $\mathcal K(s)$ mentioned after (3.20) is not clear.

Our answer: We meant that the function $\mathcal K(s)$ can be expanded as a power series of $\sqrt{-s}$, and we agree that the justification for such an expansion was not clearly stated. We have reformulated the two paragraphs after (3.21) in order to make this issue clearer.

Referee’s comment: 3) Can the KPZ scaling mentioned after (3.23) be related to a property of the spectrum of SSEP?

Our answer: we thank the referee for this interesting remark; it is true that the gap between the maximal eigenvalue of the operator and the next one has been computed in varied systems (the TASEP and the ASEP), and is found to scale as $1/L^{3/2}$ (refs [73,126,127] of the resubmitted version), when there are no Legendre parameters conjugated to the current or the activity. Extending their analysis of the gap to our problem in the presence of a bias is beyond the scope of our work, but we have added a remark in the conclusion where we had already mentioned the gap. We stress in particular that a $1/L^{3/2}$ gap in the deformed matrix M of the activity in the SSEP would imply a $3/2$ dynamical exponent for large enough $|s|$ in this model.

Referee’s comment: 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?

Our answer: We have considered systems with values of $N$ and $L$ up to $16$. To compute $D$, we used the relation $D=\lim_{t\to\infty} \frac{\langle Q^2\rangle_c}{t}$ mentioned before (3.29). This expression can be evaluated numerically from the second derivative of the SCGF, which is the maximal eigenvalue of the matrix $W$. This eigenvalue was evaluated by numerically diagonalizing $W$ with arbitrary numerical precision (allowing to take the second derivative numerically). When varying $\mu'$ from $0$, the numerically obtained value of $D$ diverges when $\mu'\to\mu'_c$. We have updated the paragraph after (3.30) in order to explain this procedure.

Referee’s comment: 5) A reference for eqs (4.18) could be useful.

We thank the referee for their careful reading of the manuscript and very positive appraisal of our work. We are glad that the referee "recommend[s] publication as is".

### List of changes

§3.1:
. In the summary on page 8, titles of items are highlighted in a better way.
. In II), the definition of "finite $\lambda$" has been made explicit.

§3.3: after (3.21), the series expansion of $\mathcal K(s)$ in powers of $\sqrt{-s}$ has been justified in the paragraph before (3.22).

§3.5: the details of the numerical procedure to evaluate D and its divergence as $\mu'\to\mu'_c$ have been made explicit

§4.2: before (4.18), references [113,114] have been added

§5: in the last paragraph of the conclusion, the discussion on the gap has been expanded, conjecturing a possible KPZ-type dynamical exponent in the SCGF of the SSEP for large-enough deviations of the activity.