**6. Summary**

The two-dimensional images of the ion-track overlapping effects on the amorphization of TiO2, the crystal phase transformation of ZrO2, and the transition of the magnetic states of CeO<sup>2</sup> were simulated by the Monte Carlo method. From the two-dimensional images, the fractions of the areas which are modified by ion-tracks were calculated as a function of ion fluence. They agree well with the analytical result by using the Poisson distribution function.

**Author Contributions:** Monte Carlo simulation, S.N.; data analysis by Poisson distribution function, A.I.; data curation, S.N.; writing—original draft preparation, A.I.; writing—review and editing, A.I. and S.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** One of the authors (A.I.) thanks N. Ishikawa and T. Yamaki for fruitful discussion.

**Conflicts of Interest:** The authors declare no conflict of interest.
