Mathematical Finance, Stochastic Analysis, and Machine Learning Seminar by Leonardo Tolomeo: Phase Transitions of the Focusing Φ^p_1 Measures

Speaker:

Leonardo Tolomeo, Hausdorff Centre for Mathematics, Bonn, Germany

Title:

Phase Transitions of the Focusing Φ^p_1 Measures

Abstract:

We study the behavior of the focusing Φ^p_1 measures on the one-dimensional torus initiated by Lebowitz, Rose, and Speer (1988). Because of the focusing nature of the measure, it is necessary to introduce a mass cutoff K, so that the measure is formally given by the expression

Z^{-1} χ(\int |φ|^2 \le K) \exp( β/p \int |φ|^p - 1/2 \int |φ|^2 - \int |φ’|^2) d φ.

We exhibit the following phase transitions:

- When p = 6, the measure is normalisable for K \le K_0( β), and it is not normalisable for K > K_0(β). The endpoint result K = K_0 answers an open question of the original paper by Lebowitz, Rose, and Speer. This is a joint work with T. Oh and P. Sosoe.

- When p < 6, we consider the limit case of a big torus with weak potential, i.e. K = LK_0, β=L^{-γ}β_0, where L is the size of the torus. We show that when γ=γ(p) is appropriate, the limiting behavior depends on the value of β_0. In particular, when β_0 is small, we show that the Φ^p_1 converges to the Ornstein–Uhlenbeck measure on the real line. When β_0 is big, the measure converges to the \delta measure concentrated in 0 (and it actually concentrates around a single soliton, appropriately rescaled). This extends a result of Rider (2002). This is a joint work with H. Weber.

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Mathematical Finance, Stochastic Analysis, and Machine Learning Seminar