# Topological effects and conformal invariance in long-range correlated random surfaces

### Submission summary

 As Contributors: Sebastian Grijalva · Nina Javerzat Arxiv Link: https://arxiv.org/abs/2005.11830v2 (pdf) Date accepted: 2020-09-17 Date submitted: 2020-08-05 16:28 Submitted by: Javerzat, Nina Submitted to: SciPost Physics Discipline: Physics Subject area: Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

We consider discrete random fractal surfaces with negative Hurst exponent $H<0$. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level $h$. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter ($H$) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value $h=h_c$ and for $H\leq-\frac{3}{4}$ the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For $-\frac{3}{4}<H< 0$ instead, there is a line of critical points with continously varying exponents. The universality class of these points, in particular concerning the conformal invariance of the level clusters, is poorly understood. By combining the Conformal Field Theory and the numerical approach, we provide new insights on these phases. We focus on the connectivity function, defined as the probability that two sites belong to the same level cluster. In our simulations, the surfaces are defined on a lattice torus of size $M\times N$. We show that the topological effects on the connectivity function make manifest the conformal invariance for all the critical line $H<0$. In particular, exploiting the anisotropy of the rectangular torus ($M\neq N$), we directly test the presence of the two components of the traceless stress-energy tensor. Moreover, we probe the spectrum and the structure constants of the underlying Conformal Field Theory. Finally, we observed that the corrections to the scaling clearly point out a breaking of integrability moving from the pure percolation point to the long-range correlated one.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### Submission & Refereeing History

Resubmission 2005.11830v2 on 5 August 2020
Submission 2005.11830v1 on 3 June 2020

## Reports on this Submission

### Report

I thank the authors for the modifications brought to their manuscript, which in my opinion make it ready for publication in SciPost.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

The authors have taken into account all my comments and I now recommend publication.

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