1,466 research outputs found
Combinatorics of lattice paths
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014.This dissertation consists of ve chapters which deal with lattice paths such as
Dyck paths, skew Dyck paths and generalized Motzkin paths. They never go below the horizontal axis. We derive the
generating functions to enumerate lattice paths according to di erent parameters.
These parameters include strings of length 2, 3, 4 and r for all r 2 f2; 3; 4; g,
area and semi-base, area and semi-length, and semi-base and semi-perimeter. The
coe cients in the series expansion of these generating functions give us the number
of combinatorial objects we are interested to count. In particular
1. Chapter 1 is an introduction, here we derive some tools that we are going to
use in the subsequent Chapters. We rst state the Lagrange inversion formula which
is a remarkable tool widely use to extract coe cients in generating functions, then
we derive some generating functions for Dyck paths, skew Dyck paths and Motzkin
paths.
2. In Chapter 2 we use generating functions to count the number of occurrences
of strings in a Dyck path. We rst derive generating functions for strings of length 2,
3, 4 and r for all r 2 f2; 3; 4; g, we then extract the coe cients in the generating
functions to get the number of occurrences of strings in the Dyck paths of semi-length
n.
3. In Chapter 3, Sections 3.1 and 3.2 we derive generating functions for the
relationship between strings of lengths 2 and 3 and the relationship between strings
of lengths 3 and 4 respectively. In Section 3.3 we derive generating functions for the
low occurrences of the strings of lengths 2, 3 and 4 and lastly Section 3.4 deals with
derivations of generating functions for the high occurrences of some strings .
4. Chapter 4, Subsection 4.1.1 deals with the derivation of the generating functions
for skew paths according to semi-base and area, we then derive the generating
functions according to area. In Subsection 4.1.2, we consider the same as in Section
4.1.1, but here instead of semi-base we use semi-length. The last section 4.2, we
use skew paths to enumerate the number of super-diagonal bar graphs according to
perimeter.
5. Chapter 5 deals with the derivation of recurrences for the moments of generalized
Motzkin paths, in particular we consider those Motzkin paths that never
touch the x-axis except at (0,0) and at the end of the path
Criticality without frustration for quantum spin-1 chains
Frustration-free (FF) spin chains have a property that their ground state
minimizes all individual terms in the chain Hamiltonian. We ask how entangled
the ground state of a FF quantum spin-s chain with nearest-neighbor
interactions can be for small values of s. While FF spin-1/2 chains are known
to have unentangled ground states, the case s=1 remains less explored. We
propose the first example of a FF translation-invariant spin-1 chain that has a
unique highly entangled ground state and exhibits some signatures of a critical
behavior. The ground state can be viewed as the uniform superposition of
balanced strings of left and right parentheses separated by empty spaces.
Entanglement entropy of one half of the chain scales as log(n)/2 + O(1), where
n is the number of spins. We prove that the energy gap above the ground state
is polynomial in 1/n. The proof relies on a new result concerning statistics of
Dyck paths which might be of independent interest.Comment: 11 pages, 2 figures. Version 2: minor changes in the proof of Lemma
Power law violation of the area law in quantum spin chains
The sub-volume scaling of the entanglement entropy with the system's size,
, has been a subject of vigorous study in the last decade [1]. The area law
provably holds for gapped one dimensional systems [2] and it was believed to be
violated by at most a factor of in physically reasonable
models such as critical systems.
In this paper, we generalize the spin model of Bravyi et al [3] to all
integer spin- chains, whereby we introduce a class of exactly solvable
models that are physical and exhibit signatures of criticality, yet violate the
area law by a power law. The proposed Hamiltonian is local and translationally
invariant in the bulk. We prove that it is frustration free and has a unique
ground state. Moreover, we prove that the energy gap scales as , where
using the theory of Brownian excursions, we prove . This rules out the
possibility of these models being described by a conformal field theory. We
analytically show that the Schmidt rank grows exponentially with and that
the half-chain entanglement entropy to the leading order scales as
(Eq. 16). Geometrically, the ground state is seen as a uniform superposition of
all colored Motzkin walks. Lastly, we introduce an external field which
allows us to remove the boundary terms yet retain the desired properties of the
model. Our techniques for obtaining the asymptotic form of the entanglement
entropy, the gap upper bound and the self-contained expositions of the
combinatorial techniques, more akin to lattice paths, may be of independent
interest.Comment: v3: 10+33 pages. In the PNAS publication, the abstract was rewritten
and title changed to "Supercritical entanglement in local systems:
Counterexample to the area law for quantum matter". The content is same
otherwise. v2: a section was added with an external field to include a model
with no boundary terms (open and closed chain). Asymptotic technique is
improved. v1:37 pages, 10 figures. Proc. Natl. Acad. Sci. USA, (Nov. 2016
Path representation of maximal parabolic Kazhdan-Lusztig polynomials
We provide simple rules for the computation of Kazhdan--Lusztig polynomials
in the maximal parabolic case. They are obtained by filling regions delimited
by paths with "Dyck strips" obeying certain rules. We compare our results with
those of Lascoux and Sch\"utzenberger.Comment: v3: fixed proof of lemma
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