889 research outputs found
Determinantal structures in the O'Connell-Yor directed random polymer model
We study the semi-discrete directed random polymer model introduced by
O'Connell and Yor. We obtain a representation for the moment generating
function of the polymer partition function in terms of a determinantal measure.
This measure is an extension of the probability measure of the eigenvalues for
the Gaussian Unitary Ensemble (GUE) in random matrix theory. To establish the
relation, we introduce another determinantal measure on larger degrees of
freedom and consider its few properties, from which the representation above
follows immediately.Comment: 45 pages, 2 figure
On the -TASEP with a random initial condition
When studying fluctuations of models in the 1D KPZ class including the ASEP
and the -TASEP, a standard approach has been to first write down a formula
for -deformed moments and constitute their generating function. This works
well for the step initial condition, but there is a difficulty for a random
initial condition (including the stationary case): in this case only the first
few moments are finite and the rest diverge. In a previous work [16], we
presented a method dealing directly with the -deformed Laplace transform of
an observable, in which the above difficulty does not appear. There the
Ramanujan's summation formula and the Cauchy determinant for the theta
functions play an important role. In this note, we give an alternative approach
for the -TASEP without using them.Comment: 20 page
Stationary correlations for the 1D KPZ equation
We study exact stationary properties of the one-dimensional
Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The
stationary state for the KPZ equation is realized by setting the initial
condition the two-sided Brownian motion (BM) with respect to the space
variable. Developing techniques for dealing with this initial condition in the
replica analysis, we elucidate some exact nature of the height fluctuation for
the KPZ equation. In particular, we obtain an explicit representation of the
probability distribution of the height in terms of the Fredholm determinants.
Furthermore from this expression, we also get the exact expression of the
space-time two-point correlation function.Comment: 38 pages, 5 figure
Entanglement generation through an open quantum dot: an exact approach
We analytically study entanglement generation through an open quantum dot
system described by the two-lead Anderson model. We exactly obtain the
transition rate between the non-entangled incident state in one lead and the
outgoing spin-singlet state in the other lead. In the cotunneling process, only
the spin-singlet state can transmit. To discuss such an entanglement property
in the open quantum system, we construct the exact two-electron scattering
state of the Anderson model. It is striking that the scattering state contains
spin-singlet bound states induced by the Coulomb interaction. The bound state
describes the scattering process in which the set of momenta is not conserved
and hence it is not in the form of a Bethe eigenstate.Comment: 5 pages, 2 figure
Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition
For the two-sided Bernoulli initial condition with density (resp.
) to the left (resp. to the right), we study the distribution of a
tagged particle in the one dimensional symmetric simple exclusion process. We
obtain a formula for the moment generating function of the associated current
in terms of a Fredholm determinant. Our arguments are based on a combination of
techniques from integrable probability which have been developed recently for
studying the asymmetric exclusion process and a subsequent intricate symmetric
limit. An expression for the large deviation function of the tagged particle
position is obtained, including the case of the stationary measure with uniform
density .Comment: 35 pages, 1 figur
Free-hand gas identification based on transfer function ratios without gas flow control
Gas identification is one of the most important functions of gas sensor
systems. To identify gas species from sensing signals, however, gas input
patterns (e.g. the gas flow sequence) must be controlled or monitored precisely
with additional instruments such as pumps or mass flow controllers; otherwise,
effective signal features for analysis are difficult to be extracted. Toward a
compact and easy-to-use gas sensor system that can identify gas species, it is
necessary to overcome such restrictions on gas input patterns. Here we develop
a novel gas identification protocol that is applicable to arbitrary gas input
patterns without controlling or monitoring any gas flow. By combining the
protocol with newly developed MEMS-based sensors (i.e. Membrane-type Surface
stress Sensors (MSS)), we have realized the gas identification with the
free-hand measurement, in which one can simply hold a small sensor chip near
samples. From sensing signals obtained through the free-hand measurement, we
have developed machine learning models that can identify not only solvent
vapors but also odors of spices and herbs with high accuracies. Since no bulky
gas flow control units are required, this protocol will expand the
applicability of gas sensors to portable electronics and wearable devices,
leading to practical artificial olfaction.Comment: 19 pages, 8 figures, 3 table
Replica approach to the KPZ equation with half Brownian motion initial condition
We consider the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with half
Brownian motion initial condition, studied previously through the weakly
asymmetric simple exclusion process. We employ the replica Bethe ansatz and
show that the generating function of the exponential moments of the height is
expressed as a Fredholm determinant. From this the height distribution and its
asymptotics are studied. Furthermore using the replica method we also discuss
the multi-point height distribution. We find that some nice properties of the
deformed Airy functions play an important role in the analysis.Comment: 37 pages, 2 figure
Solvable models in the KPZ class: approach through periodic and free boundary Schur measures
We explore probabilistic consequences of correspondences between
-Whittaker measures and periodic and free boundary Schur measures
established by the authors in the recent paper [arXiv:2106.11922]. The result
is a comprehensive theory of solvability of stochastic models in the KPZ class
where exact formulas descend from mapping to explicit determinantal and
pfaffian point processes. We discover new variants of known results as
determinantal formulas for the current distribution of the ASEP on the line and
new results such as Fredholm pfaffian formulas for the distribution of the
point-to-point partition function of the Log Gamma polymer model in half space.
In the latter case, scaling limits and asymptotic analysis allow to establish
Baik-Rains phase transition for height function of the KPZ equation on the half
line at the origin.Comment: Comments are welcom
Stationary Higher Spin Six Vertex Model and -Whittaker measure
In this paper we consider the Higher Spin Six Vertex Model on the lattice
. We first identify a family of
translation invariant measures and subsequently we study the one point
distribution of the height function for the model with certain random boundary
conditions. Exact formulas we obtain prove to be useful in order to establish
the asymptotic of the height distribution in the long space-time limit for the
stationary Higher Spin Six Vertex Model. In particular, along the
characteristic line we recover Baik-Rains fluctuations with size of
characteristic exponent . We also consider some of the main degenerations
of the Higher Spin Six Vertex Model and we adapt our analysis to the relevant
cases of the -Hahn particle process and of the Exponential Jump Model.Comment: 82 pages, 17 figures, 2 table
Mechanism underlying dynamic scaling properties observed in the contour of spreading epithelial monolayer
We found evidence of dynamic scaling in the spreading of MDCK monolayer,
which can be characterized by the Hurst exponent and the
growth exponent , and theoretically and experimentally
clarified the mechanism that governs the contour shape dynamics. During the
spreading of the monolayer, it is known that so-called "leader cells" generate
the driving force and lead the other cells. Our time-lapse observations of cell
behavior showed that these leader cells appeared at the early stage of the
spreading, and formed the monolayer protrusion. Informed by these observations,
we developed a simple mathematical model that included differences in cell
motility, cell-cell adhesion, and random cell movement. The model reproduced
the quantitative characteristics obtained from the experiment, such as the
spreading speed, the distribution of the increment, and the dynamic scaling
law. Analysis of the model equation revealed that the model could reproduce the
different scaling law from to , and the exponents were determined by the
two indices: and . Based on the analytical result, parameter
estimation from the experimental results was achieved. The monolayer on the
collagen-coated dishes showed a different scaling law , suggesting that cell motility increased by 9 folds. This result was
consistent with the assay of the single-cell motility. Our study demonstrated
that the dynamics of the contour of the monolayer were explained by the simple
model, and proposed a new mechanism that exhibits the dynamic scaling property.Comment: 11 pages, 6 figures, and supplemental material
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