The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs

Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any epsilon > 0 `most' graphs have their largest non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the absolute value of the largest non-trivial eigenvalue is at most 2 sqrt{d-1} then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, coding theory and cryptography. As many of these applications depend on the size of the largest non-trivial positive and negative eigenvalues, it is natural to investigate their distributions. We show these are well-modeled by the beta=1 Tracy-Widom distribution for several families. If the observed growth rates of the mean and standard deviation as a function of the number of vertices holds in the limit, then in the limit approximately 52% of d-regular graphs from bipartite families should be Ramanujan, and about 27% from non-bipartite families (assuming the largest positive and negative eigenvalues are independent).Comment: 23 pages, version 2 (MAJOR correction: see footnote 7 on page 7: the eigenvalue program unkowingly assumed the eigenvalues of the matrix were symmetric, which is only true for bipartite graphs; thus the second largest positive eigenvalue was returned instead of the largest non-trivial eigenvalue). To appear in Experimental Mathematic

Teaching Bank Runs with Classroom Experiments

Once relegated to cinema or history lectures, bank runs have become a modern phenomenon that captures the interest of students. We use a simple classroom experiment based upon the Diamond-Dybvig Model (1983) to demonstrate how a bank run, a seemingly irrational event, can occur rationally. We then present possible topics for discussion including various ways to prevent bank runs and moral hazard.bank runs; multiple equilibria

A New View of the Size-Mass Distribution of Galaxies: Using r20r_{20} and r80r_{80} instead of r50r_{50}

When investigating the sizes of galaxies it is standard practice to use the half-light radius, $r_{50}$. Here we explore the effects of the size definition on the distribution of galaxies in the size -- stellar mass plane. Specifically, we consider $r_{20}$ and $r_{80}$, the radii that contain 20% and 80% of a galaxy's total luminosity, as determined from a Sersic profile fit, for galaxies in the 3D-HST/CANDELS and COSMOS-DASH surveys. These radii are calculated from size catalogs based on a simple calculation assuming a Sersic profile. We find that the size-mass distributions for $r_{20}$ and $r_{80}$ are markedly different from each other and also from the canonical $r_{50}$ distribution. The most striking difference is in the relative sizes of star forming and quiescent galaxies at fixed stellar mass. Whereas quiescent galaxies are smaller than star forming galaxies in $r_{50}$, this difference nearly vanishes for $r_{80}$. By contrast, the distance between the two populations increases for $r_{20}$. Considering all galaxies in a given stellar mass and redshift bin we detect a significant bimodality in the distribution of $r_{20}$, with one peak corresponding to star forming galaxies and the other to quiescent galaxies. We suggest that different measures of the size are tracing different physical processes within galaxies; $r_{20}$ is closely related to processes controlling the star formation rate of galaxies and $r_{80}$ may be sensitive to accretion processes and the relation of galaxies with their halos.Comment: Resubmitted to ApJL after responding to referee's comments. Please also see Mowla et al. submitted today as wel

Career planning for future opportunitites

The reason my first piece of guidance is to plan your career and seek advice is because that is what worked for me. When I was in high school and thinking about what I would like to so one day, I thought it would be great to be an architect. I thought it would be fascinating to design buildings such as the amazing architecture here in Astana – buildings that would stand for many decades and that millions of people talk about. I spoke to a trusted teacher at school and he asked me to do one very important thing – research the job market for architects. What I found out caused me to change my direction and is the reason I am here today

The bias of the submillimetre galaxy population: SMGs are poor tracers of the most massive structures in the z ~ 2 Universe

It is often claimed that overdensities of (or even individual bright) submillimetre-selected galaxies (SMGs) trace the assembly of the most-massive dark matter structures in the Universe. We test this claim by performing a counts-in-cells analysis of mock SMG catalogues derived from the Bolshoi cosmological simulation to investigate how well SMG associations trace the underlying dark matter structure. We find that SMGs exhibit a relatively complex bias: some regions of high SMG overdensity are underdense in terms of dark matter mass, and some regions of high dark matter overdensity contain no SMGs. Because of their rarity, Poisson noise causes scatter in the SMG overdensity at fixed dark matter overdensity. Consequently, rich associations of less-luminous, more-abundant galaxies (i.e. Lyman-break galaxy analogues) trace the highest dark matter overdensities much better than SMGs. Even on average, SMG associations are relatively poor tracers of the most significant dark matter overdensities because of 'downsizing': at z < ~2.5, the most-massive galaxies that reside in the highest dark matter overdensities have already had their star formation quenched and are thus no longer SMGs. At a given redshift, of the 10 per cent most-massive overdensities, only ~25 per cent contain at least one SMG, and less than a few per cent contain more than one SMG.Comment: 6 pages, 3 figures, 1 table; accepted for publication in MNRAS; minor revisions from previous version, conclusions unchange