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 Polytropic 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