Zeros of Airy Function and Relaxation Process |
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Authors: | Makoto Katori and Hideki Tanemura |
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Institution: | 1.Department of Physics, Faculty of Science and Engineering,Chuo University,Tokyo,Japan;2.Department of Mathematics and Informatics, Faculty of Science,Chiba University,Chiba,Japan |
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Abstract: | One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting
between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β=2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial
configuration, it is proved that Dyson’s model with β=2 and N particles,
$\mbox {\boldmath $\mbox {\boldmath
, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel.
The Airy function
(z){\rm Ai}(z)
is an entire function with zeros all located on the negative part of the real axis ℝ. We consider Dyson’s model with β=2 starting from the first N zeros of
Ai(z){\rm Ai}(z)
, 0>a
1>⋅⋅⋅>a
N
, N≥2. In order to properly control the effect of such initial confinement of particles in the negative region of ℝ, we put the
drift term to each Brownian motion, which increases in time as a parabolic function: Y
j
(t)=X
j
(t)+t
2/4+{d
1+∑
ℓ=1
N
(1/a
ℓ
)}t,1≤j≤N, where
d1=Ai¢(0)/Ai(0)d_{1}={\rm Ai}'(0)/{\rm Ai}(0)
. We show that, as the N→∞ limit of
$\mbox {\boldmath $\mbox {\boldmath
, we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of
(z){\rm Ai}(z)
on the negative ℝ is occupied by one particle, to the stationary state
mAi\mu_{{\rm Ai}}
. The stationary state
mAi\mu_{{\rm Ai}}
is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on ℝ and in which the Tracy-Widom
distribution describes the rightmost particle position. |
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Keywords: | |
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