9,275 research outputs found
Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit
We prove that the unique entropy solution to a scalar nonlinear conservation
law with strictly monotone velocity and nonnegative initial condition can be
rigorously obtained as the large particle limit of a microscopic
follow-the-leader type model, which is interpreted as the discrete Lagrangian
approximation of the nonlinear scalar conservation law. More precisely, we
prove that the empirical measure (respectively the discretised density)
obtained from the follow-the-leader system converges in the 1-Wasserstein
topology (respectively in ) to the unique Kruzkov entropy solution
of the conservation law. The initial data are taken in ,
nonnegative, and with compact support, hence we are able to handle densities
with vacuum. Our result holds for a reasonably general class of velocity maps
(including all the relevant examples in the applications, e.g. in the
Lighthill-Whitham-Richards model for traffic flow) with possible degenerate
slope near the vacuum state. The proof of the result is based on discrete BV
estimates and on a discrete version of the one-sided Oleinik-type condition. In
particular, we prove that the regularizing effect
for nonlinear scalar conservation laws is intrinsic of the discrete model
Integrability of the quantum KdV equation at c = -2
We present a simple a direct proof of the complete integrability of the
quantum KdV equation at , with an explicit description of all the
conservation laws.Comment: 9 page
Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation
with reflecting boundary conditions which is relevant to the Temperley--Lieb
model of loops on a strip. By use of integral formulae we prove conjectures
relating it to the weighted enumeration of Cyclically Symmetric Transpose
Complement Plane Partitions and related combinatorial objects
The solution of the quantum T-system for arbitrary boundary
We solve the quantum version of the -system by use of quantum
networks. The system is interpreted as a particular set of mutations of a
suitable (infinite-rank) quantum cluster algebra, and Laurent positivity
follows from our solution. As an application we re-derive the corresponding
quantum network solution to the quantum -system and generalize it to
the fully non-commutative case. We give the relation between the quantum
-system and the quantum lattice Liouville equation, which is the quantized
-system.Comment: 24 pages, 18 figure
Entanglement Entropy of the Low-Lying Excited States and Critical Properties of an Exactly Solvable Two-Leg Spin Ladder with Three-Spin Interactions
In this work, we investigate an exactly solvable two-leg spin ladder with
three-spin interactions. We obtain analytically the finite-size corrections of
the low-lying energies and determine the central charge as well as the scaling
dimensions. The model considered in this work has the same universality class
of critical behavior of the XX chain with central charge c=1. By using the
correlation matrix method, we also study the finite-size corrections of the
Renyi entropy of the ground state and of the excited states. Our results are in
agreement with the predictions of the conformal field theory.Comment: 10 pages, 6 figures, 2 table
The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics
A fascinating conjectural connection between statistical mechanics and
combinatorics has in the past five years led to the publication of a number of
papers in various areas, including stochastic processes, solvable lattice
models and supersymmetry. This connection, known as the Razumov-Stroganov
conjecture, expresses eigenstates of physical systems in terms of objects known
from combinatorics, which is the mathematical theory of counting. This note
intends to explain this connection in light of the recent papers by Zinn-Justin
and Di Francesco.Comment: 6 pages, 4 figures, JSTAT News & Perspective
Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries
We propose new conjectures relating sum rules for the polynomial solution of
the qKZ equation with open (reflecting) boundaries as a function of the quantum
parameter and the -enumeration of Plane Partitions with specific
symmetries, with . We also find a conjectural relation \`a la
Razumov-Stroganov between the limit of the qKZ solution and refined
numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision
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