# Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model

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## Abstract

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## 1. Introduction

## 2. Method

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The generalized trail model. Weights ${\tau}_{C}$ and ${\tau}_{X}$ are introduced for collisions and crossings at a site, respectively, and weights p and $\omega $ are introduced to model chain stiffness and attraction with a surface (dotted line).

**Figure 2.**Schematic bulk phase diagram in the $\tau $, K plane. K is the step fugacity, which controls the average length of the walk, and $\tau $ is the Boltzmann weight for the monomer-monomer interactions.

**Figure 3.**Zeros of the canonical partition function for self-avoiding trails for $N=500$ calculated for four different simulations. If we label the leading zero (the one closest to the positive real axis) ${\tau}_{0}$, the x-axis gives the real part of the zero and the y-axis the imaginary part. The complex zeros can clearly be seen to pinch the real axis near ${\tau}^{\star}=3.0$, where in the thermodynamic limit the zero will sit on the axis and denote the critical point.

**Figure 4.**Zeros of the canonical partition function for self-avoiding trails for $N=650$. Again, it can clearly be seen that the leading complex zeros pinch near ${\omega}^{\star}=2.45$, giving the estimate of the critical adsorption point.

**Figure 5.**Fisher zeros for the canonical partition function of the interacting self-avoiding trail model for sizes from $N=30$ to $N=500$ fitted with a quartic function, showing the extrapolation to the critical interaction strength $\tau =3.01\pm 0.01$ to compare with the exact value of ${\tau}_{c}=3$.

**Figure 6.**The effective bulk cross-over exponent for the trails model, ${\varphi}_{\mathrm{eff}}$ plotted against $1/N$ where the exponent is calculated using two lattice sizes. Due to parity effects, we have looked at two cases: $N1=N,{N}_{2}=N-2$ and ${N}_{1}=N,{N}_{2}=N-4$.

**Figure 7.**$\mathrm{log}\mathrm{Im}\left({\tau}_{c}\right)$ vs. $\mathrm{log}N$ for Trails $O\left(n\right)$. The slope of the best fit gives an estimate of $\varphi =0.83\pm 0.02$.

**Figure 8.**A cubic fit of the $\mathrm{Re}\left({\omega}_{c}\right)$ against $\mathrm{Im}\left({\omega}_{c}\right)$ for the surface interaction of Trails at the special surface transition with $\tau ={\tau}^{\star}=3$.

**Figure 9.**Fisher zeros for the canonical partition function of the VISAW model fitted with a quartic function, showing the extrapolation to the critical interaction strength $\tau =4.76\pm 0.04$ to compare with the value of ${\tau}_{c}=4.69\pm 0.01$ calculate from Transfer Matrices and DMRG.

**Figure 10.**$\mathrm{log}\mathrm{Im}\left({\tau}_{c}\right)$ vs. $\mathrm{log}N$ for the VISAW model. The red line gives the slope of best fit, giving an estimate of $\varphi =0.89\pm 0.01$. The points appear to lie on a straight line, and the blue line (of slope $0.83\pm 0.02$, as found for the Trails) is shown as reference. While it is possible that the line evolves further to reach the slope of the blue line, we would argue that this supports the hypothesis that the ISAT and the VISAW are in different universality classes.

**Figure 11.**Fisher zeros for the canonical partition function of the Integrable VISAW model at the integrable point fitted with a quartic function, showing the extrapolation to the critical interaction strength $\tau =2.64\pm 0.02$ to compare with the exact value of ${\tau}_{c}=2.630986\cdots $.

**Figure 12.**$\mathrm{log}\mathrm{Im}\left({\tau}_{c}\right)$ vs. $\mathrm{log}N$ for the Integrable VISAW model. The slope of the best fit gives an estimate of $\varphi =0.73\pm 0.01$.

**Figure 13.**A cubic fit of the $\mathrm{Re}\left({\omega}^{\star}\right)$ against $\mathrm{Im}\left({\omega}^{\star}\right)$ for the surface interaction of the Integrable VISAW model at the collapse transition with $p={p}^{\star}$ and $\tau ={\tau}^{\star}$.

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**MDPI and ACS Style**

Foster, D.; Kenna, R.; Pinettes, C.
Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model. *Entropy* **2019**, *21*, 153.
https://doi.org/10.3390/e21020153

**AMA Style**

Foster D, Kenna R, Pinettes C.
Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model. *Entropy*. 2019; 21(2):153.
https://doi.org/10.3390/e21020153

**Chicago/Turabian Style**

Foster, Damien, Ralph Kenna, and Claire Pinettes.
2019. "Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model" *Entropy* 21, no. 2: 153.
https://doi.org/10.3390/e21020153