Access to Opportunity Project: Final Report

This project’s goal is to lift up promising approaches, suggest new strategies and encourage honest conversations that result in public policy solutions to income and racial segregation and poverty. The overarching question that motivates this work is: What are effective policies and strategies that promote access to high-opportunity amenities for low-income families? As a first step, the researchers surveyed efforts on the ground in the metropolitan areas encompassing Seattle, Washington; Portland, Oregon; and San Diego, California, to determine whether there were any candidates for deeper study. We selected these three metropolitan areas for several reasons. First, prior interaction revealed that attention had been given to this question and that parties in each had embarked on purposeful efforts to make progress. Second, they represent a diverse array of communities that vary in significant ways, including along key economic, demographic, and social dimensions, and in some regards are bellwethers for changes beginning to take place in many parts of the country. As a consequence, experiences and successes in these places could potentially be applied to a diverse set of other urban areas across the United States. The three regions are among the largest in the United States, with Seattle and Portland being the largest in their respective states and San Diego third in California (behind Los Angeles and the Bay Area). Despite their size, they differ in important ways that result in different social and political dynamics prevailing in each location. In considering access to opportunity, one must understand the opportunities that are available in order to tailor skill-building efforts and investments in “connective infrastructure,” such as mass transit and suburban affordable housing, so that they are maximally effective. From an economic perspective, the three regions are quite different, which means that the approaches observed across the regions will potentially vary in measurable ways. In each metropolitan area, we sought the counsel of key governmental, practitioner, academic, and philanthropic players. During the course of our initial visits to each region, we met with and interviewed almost 80 people—28 in Seattle, 26 in Portland, and 24 in San Diego. Through these conversations, we identified 27 projects—nine in each metropolitan area—as being promising examples of cases where lower-income families may have achieved increased access to high-opportunity amenities. Given time, available funding, and the presence of partners willing to support our research effort by providing access to program data and program participants, we chose three projects for examination: • The San Diego Housing Commission’s Achievement Academy • Seattle/King County’s A Regional Coalition for Housing (ARCH) • Humboldt Gardens in Northeast Portlan

Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets

We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in the definition of $W({G},q)$ resulting from the fact that at certain special points $q_s$, the following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then present exact calculations of $W({G},q)$ and determine the corresponding analytic structure in the complex $q$ plane for a number of families of graphs ${G}$, including circuits, wheels, biwheels, bipyramids, and (cyclic and twisted) ladders. We study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs, all but a finite number of the zeros lie exactly on a unit circle, whose position depends on the family. Using the connection of $P(G,q)$ with the zero-temperature Potts antiferromagnet, we derive a theorem concerning the maximal finite real point of non-analyticity in $W({G},q)$, denoted $q_c$ and apply this theorem to deduce that $q_c(sq)=3$ and $q_c(hc) = (3+\sqrt{5})/2$ for the square and honeycomb lattices. Finally, numerical calculations of $W(hc,q)$ and $W(sq,q)$ are presented and compared with series expansions and bounds.Comment: 33 pages, Latex, 5 postscript figures, published version; includes further comments on large-q serie

Lower Bounds and Series for the Ground State Entropy of the Potts Antiferromagnet on Archimedean Lattices and their Duals

We prove a general rigorous lower bound for $W(\Lambda,q)=\exp(S_0(\Lambda,q)/k_B)$, the exponent of the ground state entropy of the $q$-state Potts antiferromagnet, on an arbitrary Archimedean lattice $\Lambda$. We calculate large-$q$ series expansions for the exact $W_r(\Lambda,q)=q^{-1}W(\Lambda,q)$ and compare these with our lower bounds on this function on the various Archimedean lattices. It is shown that the lower bounds coincide with a number of terms in the large-$q$ expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions $W_r(\Lambda,q)$ for large $q$ on the various lattices $\Lambda$. Plots of $W_r(\Lambda,q)$ are given, and the general dependence on lattice coordination number is noted. Lower bounds and series are also presented for the duals of Archimedean lattices. As part of the study, the chromatic number is determined for all Archimedean lattices and their duals. Finally, we report calculations of chromatic zeros for several lattices; these provide further support for our earlier conjecture that a sufficient condition for $W_r(\Lambda,q)$ to be analytic at $1/q=0$ is that $\Lambda$ is a regular lattice.Comment: 39 pages, Revtex, 9 encapsulated postscript figures, to appear in Phys. Rev.

Spanning forests and the q-state Potts model in the limit q \to 0

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal versio

Exact Potts Model Partition Functions on Ladder Graphs

We present exact calculations of the partition function $Z$ of the $q$-state Potts model and its generalization to real $q$, the random cluster model, for arbitrary temperature on $n$-vertex ladder graphs with free, cyclic, and M\"obius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed.Comment: 73 pages, Latex, 20 postscript figures, Physica A, in pres

A Little Statistical Mechanics for the Graph Theorist

In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches. This paper is an elementary general audience survey, intended to popularize the area and provide an accessible first point of entry for further exploration.Comment: 30 pages, 3 figure

Determining relative bulk viscosity of kilometre-scale crustal units using field observations and numerical modelling

Though the rheology of kilometre-scale polymineralic rock units is crucial for reliable large-scale, geotectonic models, this information is difficult to obtain. In geotectonic models, a layer is defined as an entity at the kilometre scale, even though it is heterogeneous at the millimetre to metre scale. Here, we use the shape characteristics of the boundaries between rock units to derive the relative bulk viscosity of those units at the kilometre scale. We examine the shape of a vertically oriented ultramafic, harzburgitic-lherzolitic unit, which developed a kilometre-scale pinch and swell structure at mid-crustal conditions (~ 600 °C, ~ 8.5 kbar), in the Anita Shear Zone, New Zealand. The ultramafic layer is embedded between a typical polymineralic paragneiss to the west, and a feldspar-quartz-hornblende orthogneiss, to the east. Notably, the boundaries on either side of the ultramafic layer give the ultramafics an asymmetric shape. Microstructural analysis shows that deformation was dominated by dislocation creep (n = 3). Based on the inferred rheological behaviour from the field, a series of numerical simulations are performed. Relative and absolute values are derived for bulk viscosity of the rock units by comparing boundary tortuosity difference measured on the field example and the numerical series. Our analysis shows that during deformation at mid-crustal conditions, paragneisses can be ~ 30 times less viscous than an ultramafic unit, whereas orthogneisses have intermediate viscosity, ~ 3 times greater than the paragneisses. If we assume a strain rate of 10⁻ ¹⁴ s⁻ ¹ the ultramafic, orthogneiss and paragneiss have syn-deformational viscosities of 3 × 10²², 2.3 × 10²¹ and 9.4 × 10²⁰ Pa s, respectively. Our study shows pinch and swell structures are useful as a gauge to assess relative bulk viscosity of rock units based on shape characteristics at the kilometre scale and in non-Newtonian flow regimes, even where heterogeneity occurs within the units at the outcrop scale