Abstract: I will present examples of Limit Shapes – the most probable macroscopic shape
with sharp boundaries separating frozen and fluctuating regions – which arise
in a variety of classical and quantum systems. I will explain a special role
played by analytic functions defining Riemann surface whose topology can be
changed abruptly across phase transitions. Most of the examples are based on
free fermionic models, however recently we studied a notable exception from
this rule – the Polytropic Gas with a power-law equation of state. The hydrodynamic
approach to Emptiness Formation Probability in Politropic Gas will be discussed.

[1] J. Pallister, D.M. Gangardt and A. Abanov, « Limit shape phase transitions: a merger of arctic circles »,
J. Phys. A 55, 304001, 2022

[2] James S. Pallister, Samuel H. Pickering, Dimitri M. Gangardt and Alexander G. Abanov, « Phase transitions in full counting statistics of free fermions and directed polymers », Phys. Rev. Research 7, L022008, 2025

[3] Alexander G. Abanov, Dimitri M. Gangardt, « Emptiness instanton in quantum polytropic gas », SciPost Phys. 18, 122, 2025