Within the province of condensed matter physics, there exists a variety of interesting physical phenomena where we are concerned with the statistical fluctuations exhibited by an essentially linear elastic object, such as a hydrophilic polymer chain wafting in water. Due to the thermal fluctuation, the shape of the polymer chain should be understood as a random path. The water in this physical system plays the role of the disordered environment that contains randomly placed hydrophobic molecules as impurities, which repel the hydrophilic monomers that the polymer chain consists of. This physical system is called the directed polymer model. The major problem about this system is to investigate the behavior of the polymer chain for various disorder strengths. 

 

In this talk, we will focus on the -dimensional directed polymer model when . In this case, the system is quite sensitive to the disorder strength. Moreover, it is widely believed that the model has a phase transition when the disorder strength is strong enough. Consequently, it is crucial to investigate the behavior of system until a critical disorder strength that the system may have a phase transition. As a result, the goal of this talk is to present the results given by D. Lygkonis and N. Zygouras [1], which studied the limiting fluctuations of the partition function and the free energy of the directed polymer model when the disorder strength is less than a critical value. 

 

[1] D. Lygkonis and N. Zygouras. Edwards–Wilkinson fluctuations for the directed polymer in the full L²-regime for dimensions d ≥ 3. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 58(1):65 – 104, 2022. doi:10.1214/21-AIHP1173.

 

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