Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small correction

Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of "admissible" trees with possibly doubly-occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde

Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Vacancy localization in the square dimer model

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57/4-10Sqrt[2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9/8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1/4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.Comment: 35 pages, 24 figures. Improved version with one added figure (figure 9), a shift s->s+1 in the definition of the tree size, and minor correction

Molecular Aspects of HTLV-1 Entry: Functional Domains of the HTLV-1 Surface Subunit (SU) and Their Relationships to the Entry Receptors

The initial step in retroviral infection involves specific interactions between viral envelope proteins (Env) and specific receptors on the surface of target cells. For many years, little was known about the entry receptors for HTLV-1. During this time, however, functional domains of the HTLV-1 Env were identified by analyzing the effects of neutralizing antibodies and specific mutations in Env on HTLV-1 infectivity. More recent studies have revealed that HTLV-1 infectivity involves interactions with three different molecules: heparan sulfate proteoglycans (HSPG), the VEGF-165 receptor Neuropilin 1 (NRP-1) and glucose transporter type 1 (GLUT1). Here, we revisit previously published data on the functional domains of Env in regard to the recent knowledge acquired about this multi-receptor complex. We also discuss the similarities and differences between HTLV-1 and other deltaretroviruses in regards to receptor usage

Coupling of hard dimers to dynamical lattices via random tensors

We study hard dimers on dynamical lattices in arbitrary dimensions using a random tensor model. The set of lattices corresponds to triangulations of the d-sphere and is selected by the large N limit. For small enough dimer activities, the critical behavior of the continuum limit is the one of pure random lattices. We find a negative critical activity where the universality class is changed as dimers become critical, in a very similar way hard dimers exhibit a Yang-Lee singularity on planar dynamical graphs. Critical exponents are calculated exactly. An alternative description as a system of `color-sensitive hard-core dimers' on random branched polymers is provided.Comment: 12 page

Mechanism of Splicing Regulation of Spinal Muscular Atrophy Genes

Spinal muscular atrophy (SMA) is one of the major genetic disorders associated with infant mortality. More than 90% cases of SMA result from deletions or mutations of Survival Motor Neuron 1 (SMN1) gene. SMN2, a nearly identical copy of SMN1, does not compensate for the loss of SMN1due to predominant skipping of exon 7. However, correction of SMN2 exon 7 splicing has proven to confer therapeutic benefits in SMA patients. The only approved drug for SMA is an antisense oligonucleotide (Spinraza™/Nusinersen), which corrects SMN2 exon 7 splicing by blocking intronic splicing silencer N1 (ISS-N1) located immediately downstream of exon 7. ISS-N1 is a complex regulatory element encompassing overlapping negative motifs and sequestering a cryptic splice site. More than 40 protein factors have been implicated in the regulation of SMN exon 7 splicing. There is evidence to support that multiple exons of SMN are alternatively spliced during oxidative stress, which is associated with a growing number of pathological conditions. Here, we provide the most up to date account of the mechanism of splicing regulation of the SMN genes

Trees and spatial topology change in causal dynamical triangulations

Contains fulltext : 117305.pdf (preprint version ) (Open Access

Embedded Trees and the Support of the ISE

Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by ±1. Recently it was proved (see [6] and [9]) that the distribution of the maximum and minimum label are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier, Di Francesco, and Guitter [5] that is based on properties of Jacobi’s θ-functions. In particular we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass ℘-function. Furthermore we re-derive the limiting radius distribution in random quadrangulations (by Chassaing and Schaeffer [6]) with the help of exact counting generating functions.