d-Path Laplacians and Quantum Transport on Graphs

We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph-a graph consisting of two cliques separated by a path-the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian

From Quantum Graph Computing to Quantum Graph Learning: A Survey

Quantum computing (QC) is a new computational paradigm whose foundations relate to quantum physics. Notable progress has been made, driving the birth of a series of quantum-based algorithms that take advantage of quantum computational power. In this paper, we provide a targeted survey of the development of QC for graph-related tasks. We first elaborate the correlations between quantum mechanics and graph theory to show that quantum computers are able to generate useful solutions that can not be produced by classical systems efficiently for some problems related to graphs. For its practicability and wide-applicability, we give a brief review of typical graph learning techniques designed for various tasks. Inspired by these powerful methods, we note that advanced quantum algorithms have been proposed for characterizing the graph structures. We give a snapshot of quantum graph learning where expectations serve as a catalyst for subsequent research. We further discuss the challenges of using quantum algorithms in graph learning, and future directions towards more flexible and versatile quantum graph learning solvers

Localization for Linearly Edge Reinforced Random Walks

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.Comment: 30 page

Stationary random metrics on hierarchical graphs via (min⁡,+)(\min,+)-type recursive distributional equations

This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how to define a stationary random metric on self-similar spaces which are the limit of nice finite graphs: these are the so-called hierarchical graphs. They possess a well-defined level structure and any level is built using a simple recursion. Stopping the construction at any finite level, we have a discrete random metric space when we set the edges to have random length (using a multiplicative cascade with fixed law $m$). We introduce a tool, the cut-off process, by means of which one finds that renormalizing the sequence of metrics by an exponential factor, they converge in law to a non-trivial metric on the limit space. Such limit law is stationary, in the sense that glueing together a certain number of copies of the random limit space, according to the combinatorics of the brick graph, the obtained random metric has the same law when rescaled by a random factor of law $m$. In other words, the stationary random metric is the solution of a distributional equation. When the measure $m$ has continuous positive density on $\mathbf{R}_+$, the stationary law is unique up to rescaling and any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation. We also investigate topological and geometric properties of the random space when $m$ is $\log$-normal, detecting a phase transition influenced by the branching random walk associated to the multiplicative cascade.Comment: 75 pages, 16 figures. This is a substantial improvement of the first version: title changed (formerly "Quantum gravity and (min,+)-type recursive distributional equations"), the presentation has been restyled and new main results adde

DIMERS AND IMAGINARY GEOMETRY

We present a general result which shows that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds true assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. This includes the case of lozenge tilings with boundary conditions lying in a plane, and Temperleyan domains in isoradial graphs (recovering a recent result of Li). The robustness of our approach, which is a key novelty of this paper, comes from the fact that the exactly solvable nature of the model plays only a minor role in the analysis. Instead, we rely on a connection to imaginary geometry, where the limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field