36 research outputs found
Non-intersecting squared Bessel paths: critical time and double scaling limit
We consider the double scaling limit for a model of non-intersecting
squared Bessel processes in the confluent case: all paths start at time
at the same positive value , remain positive, and are conditioned to end
at time at . After appropriate rescaling, the paths fill a region in
the --plane as that intersects the hard edge at at a
critical time . In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as of the correlation kernel at critical time and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.Comment: 53 pages, 15 figure
Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
We study a model of non-intersecting squared Bessel processes in the
confluent case: all paths start at time at the same positive value , remain positive, and are conditioned to end at time at . In
the limit , after appropriate rescaling, the paths fill out a
region in the -plane that we describe explicitly. In particular, the paths
initially stay away from the hard edge at , but at a certain critical
time the smallest paths hit the hard edge and from then on are stuck to
it. For we obtain the usual scaling limits from random matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that the
positions of the paths at any time constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-type
weights. As a consequence, there is a matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in the large
limit using the Deift-Zhou steepest descent method. There are some novel
ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure
System of Complex Brownian Motions Associated with the O'Connell Process
The O'Connell process is a softened version (a geometric lifting with a
parameter ) of the noncolliding Brownian motion such that neighboring
particles can change the order of positions in one dimension within the
characteristic length . This process is not determinantal. Under a special
entrance law, however, Borodin and Corwin gave a Fredholm determinant
expression for the expectation of an observable, which is a softening of an
indicator of a particle position. We rewrite their integral kernel to a form
similar to the correlation kernels of determinantal processes and show, if the
number of particles is , the rank of the matrix of the Fredholm determinant
is . Then we give a representation for the quantity by using an -particle
system of complex Brownian motions (CBMs). The complex function, which gives
the determinantal expression to the weight of CBM paths, is not entire, but in
the combinatorial limit it becomes an entire function providing
conformal martingales and the CBM representation for the noncolliding Brownian
motion is recovered.Comment: v3: AMS_LaTeX, 25 pages, no figure, minor corrections made for
publication in J. Stat. Phy
Determinantal process starting from an orthogonal symmetry is a Pfaffian process
When the number of particles is finite, the noncolliding Brownian motion
(BM) and the noncolliding squared Bessel process with index
(BESQ) are determinantal processes for arbitrary fixed initial
configurations. In the present paper we prove that, if initial configurations
are distributed with orthogonal symmetry, they are Pfaffian processes in the
sense that any multitime correlation functions are expressed by Pfaffians. The
skew-symmetric matrix-valued correlation kernels of the Pfaffians
processes are explicitly obtained by the equivalence between the noncolliding
BM and an appropriate dilatation of a time reversal of the temporally
inhomogeneous version of noncolliding BM with finite duration in which all
particles start from the origin, , and by the equivalence between
the noncolliding BESQ and that of the noncolliding squared
generalized meander starting from .Comment: v2: AMS-LaTeX, 17 pages, no figure, corrections made for publication
in J.Stat.Phy
Noncolliding Squared Bessel Processes
We consider a particle system of the squared Bessel processes with index conditioned never to collide with each other, in which if
the origin is assumed to be reflecting. When the number of particles is finite,
we prove for any fixed initial configuration that this noncolliding diffusion
process is determinantal in the sense that any multitime correlation function
is given by a determinant with a continuous kernel called the correlation
kernel. When the number of particles is infinite, we give sufficient conditions
for initial configurations so that the system is well defined. There the
process with an infinite number of particles is determinantal and the
correlation kernel is expressed using an entire function represented by the
Weierstrass canonical product, whose zeros on the positive part of the real
axis are given by the particle-positions in the initial configuration. From the
class of infinite-particle initial configurations satisfying our conditions, we
report one example in detail, which is a fixed configuration such that every
point of the square of positive zero of the Bessel function is
occupied by one particle. The process starting from this initial configuration
shows a relaxation phenomenon converging to the stationary process, which is
determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in
J. Stat. Phy
Bessel Process and Conformal Quantum Mechanics
Different aspects of the connection between the Bessel process and the
conformal quantum mechanics (CQM) are discussed. The meaning of the possible
generalizations of both models is investigated with respect to the other model,
including self adjoint extension of the CQM. Some other generalizations such as
the Bessel process in the wide sense and radial Ornstein- Uhlenbeck process are
discussed with respect to the underlying conformal group structure.Comment: 28 Page
Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition
For the symmetric simple exclusion process on an infinite line, we calculate
exactly the fluctuations of the integrated current during time
through the origin when, in the initial condition, the sites are occupied with
density on the negative axis and with density on the positive
axis. All the cumulants of grow like . In the range where , the decay of the distribution of is
non-Gaussian. Our results are obtained using the Bethe ansatz and several
identities recently derived by Tracy and Widom for exclusion processes on the
infinite line.Comment: 2 figure
Exact solution of the six-vertex model with domain wall boundary conditions. Antiferroelectric phase
We obtain the large asymptotics of the partition function of the
six-vertex model with domain wall boundary conditions in the antiferroelectric
phase region, with the weights a=\sinh(\ga-t), b=\sinh(\ga+t), c=\sinh(2\ga),
|t|<\ga. We prove the conjecture of Zinn-Justin, that as ,
Z_n=C\th_4(n\om) F^{n^2}[1+O(n^{-1})], where \om and are given by
explicit expressions in \ga and , and is the Jacobi theta
function. The proof is based on the Riemann-Hilbert approach to the large
asymptotic expansion of the underlying discrete orthogonal polynomials and on
the Deift-Zhou nonlinear steepest descent method.Comment: 69 pages, 10 figure
The hypothetical Old-Northern chromosome race of Sorex araneus found in the Ural Mts
Available from JSOR at https://www.jstor.org/stable/23735690Chromosomes of two populations of the common shrew, Sorex araneus L. (Mammalia, Insectivora, Soricidae), from the northern Ural Mts. were investigated. In both sites, homozygous, all-metacentric autosomal complements were revealed, with the autosomal arm combinations af, bc, go, hn, ip, jl, km, qr, tu. This karyotype is identical to that predicted by Halkka et al. (1994) as the hypothetical Old-Northern race connecting the northern and eastern ratial groups of Sorex araneus in Eurasia.This study was supported by grants from the INTAS (No. 93-1463), Russian Foundation of Fundamental Research (No. 95-04-12698a), and Grant Agency of the Academy of Sciences of the Czech Republic (No. A6045601)
Chromosome synapsis and recombination in male hybrids between two chromosome races of the common shrew (Sorex araneus L., Soricidae, Eulipotyphla)
Hybrid zones between chromosome races of the common shrew (Sorex araneus) provide exceptional models to study the potential role of chromosome rearrangements in the initial steps of speciation. The Novosibirsk and Tomsk races differ by a series of Robertsonian fusions with monobrachial homology. They form a narrow hybrid zone and generate hybrids with both simple (chain of three chromosomes) and complex (chain of eight or nine) synaptic configurations. Using immunolocalisation of the meiotic proteins, we examined chromosome pairing and recombination in males from the hybrid zone. Homozygotes and simple heterozygotes for Robertsonian fusions showed a low frequency of synaptic aberrations (<10%). The carriers of complex synaptic configurations showed multiple pairing abnormalities, which might lead to reduced fertility. The recombination frequency in the proximal regions of most chromosomes of all karyotypes was much lower than in the other regions. The strong suppression of recombination in the pericentromeric regions and co-segregation of race specific chromosomes involved in the long chains would be expected to lead to linkage disequilibrium between genes located there. Genic differentiation, together with the high frequency of pairing aberrations in male carriers of the long chains, might contribute to maintenance of the narrow hybrid zone.This work was supported by INTAS (Grant # 03-51-4030) for J.B. Searle, The Russian Foundation for Basic Research (Grant # 16-04-00087) for P.M. Borodin and The Federal Agency for Scientific Organizations (Grant # 0324-2016-0024) for all authors of this paper affiliated with the Institute of Cytology and Genetics of the Siberian Department of the Russian Academy of Sciences