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Basic properties of critical lognormal multiplicative chaos. ArXiv eprints
, 2013
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Generalized Random Energy Model at complex temperatures
, 2014
"... Abstract. Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature β. We compute the limiting logpartition function and describe the fluctuations of the partition function. For the GREM with ..."
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Abstract. Motivated by the Lee–Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature β. We compute the limiting logpartition function and describe the fluctuations of the partition function. For the GREM with d levels, in total, there are 1 2 (d+ 1)(d+ 2) phases, each of which can symbolically be encoded as Gd1F d2Ed3 with d1, d2, d3 ∈ N0 such that d1 + d2 + d3 = d. In phase Gd1F d2Ed3, the first d1 levels (counting from the root of the GREM tree) are in the glassy phase (G), the next d2 levels are dominated by fluctuations (F), and the last d3 levels are dominated by the expectation (E). Only the phases of the form Gd1Ed3 intersect the real β axis. We describe the limiting distribution of the zeros of the partition function in the complex β plane ( = Fisher zeros). It turns out that the complex zeros densely touch the positive real axis at d points at which the GREM is known to undergo phase transitions. Our results confirm rigorously and considerably extend the replicamethod predictions from the physics literature. Figure 1. Phase diagram of the GREM in the complex β plane together with the level lines of the limiting logpartition function. See Figure 4 for details.
The glassy phase of complex branching Brownian motion
, 2013
"... In this paper, we study complex valued branching Brownian motion in the socalled glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on com ..."
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In this paper, we study complex valued branching Brownian motion in the socalled glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so called derivative martingale, mainly due to the clustering of the modulus. The proof relies on the fine description of the extremal process available in the branching Brownian
Tightness of the recentered maximum of logcorrelated Gaussian fields
, 2013
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THE CHARACTERISTIC POLYNOMIAL OF A RANDOM UNITARY MATRIX AND GAUSSIAN MULTIPLICATIVE CHAOS THE L2PHASE.
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