30,274 research outputs found
Multipoint correlators in the Abelian sandpile model
We revisit the calculation of height correlations in the two-dimensional
Abelian sandpile model by taking advantage of a technique developed recently by
Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian,
ubiquitous in the context of cycle-rooted spanning forests, with a complex
connection. In the case at hand, the connection is constant and localized along
a semi-infinite defect line (zipper). In the appropriate limit of a trivial
connection, it allows one to count spanning forests whose components contain
prescribed sites, which are of direct relevance for height correlations in the
sandpile model. Using this technique, we first rederive known 1- and 2-site
lattice correlators on the plane and upper half-plane, more efficiently than
what has been done so far. We also compute explicitly the (new) next-to-leading
order in the distances ( for 1-site on the upper half-plane,
for 2-site on the plane). We extend these results by computing new correlators
involving one arbitrary height and a few heights 1 on the plane and upper
half-plane, for the open and closed boundary conditions. We examine our lattice
results from the conformal point of view, and confirm the full consistency with
the specific features currently conjectured to be present in the associated
logarithmic conformal field theory.Comment: 60 pages, 21 figures. v2: reformulation of the grove theorem, minor
correction
Maximal families of nodal varieties with defect
In this paper we prove that a nodal hypersurface in P^4 with defect has at
least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it
contains either a plane or a quadric surface. Furthermore, we prove that a
nodal double cover of P^3 ramified along a surface of degree 2d with defect has
at least d(2d-1) nodes. We construct the largest dimensional family of nodal
degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large.Comment: v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section
5); Some minor corrections in the other sections. v3: some minor corrections
in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has
been modified; The paper is split into two parts, the complete intersection
case will be discussed in a different pape
Extensions of rich words
In [X. Droubay et al, Episturmian words and some constructions of de Luca and
Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has
at most |w|+1 many distinct palindromic factors, including the empty word. The
unified study of words which achieve this limit was initiated in [A. Glen et
al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these
words rich (in palindromes).
This article contains several results about rich words and especially
extending them. We say that a rich word w can be extended richly with a word u
if wu is rich. Some notions are also made about the infinite defect of a word,
the number of rich words of length n and two-dimensional rich words.Comment: 19 pages, 3 figure
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