Peter G. Doyle

Random walks and electric networks

Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. In this work we will look at the interplay of physics and mathematics in terms of an example where the mathematics involved is at the college level. The example is the relation between elementary electric network theory and random walks. Central to the work will be Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the starting point when d ≥ 3. Our goal will be to interpret this theorem as a statement about electric networks, and then to prove the theorem using techniques from classical electrical theory. The techniques referred to go back to Lord Rayleigh, who introduced them in connection with an investigation of musical instruments. The analog of Polya’s theorem in this connection is that wind instruments are possible in our three-dimensional world, but are not possible in Flatland (Abbott [1]). The connection between random walks and electric networks has been recognized for some time (see Kakutani [12], Kemeny, Snell, and

Random walks on weighted graphs and applications to on-line algorithms

We study the design and analysis of randomized on-line algorithms. We show that this problem is closely related to the synthesis of random walks on graphs with positive real costs on their edges.

Solving the Quintic by Iteration

Equations that can be solved using iterated rational maps are characterized: an equation is ‘computable’ if and only if its Galois group is within A5 of solvable. We give explicitly a new solution to the quintic polynomial, in which the transcendental inversion of the icosahedral map (due to Hermite and Kronecker) is replaced by a purely iterative algorithm. The algorithm requires a rational map with icosahedral symmetries; we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials.

Random walks on weighted graphs, and applications to on-line algorithms

We study the design and analysis of randomized on-line algorithms. We show that this problem is closely related to the synthesis of random walks on graphs with positive real costs on their edges. We develop a theory for the synthesis of such walks, and employ it to design competitive on-line algorithms. IBM T.J. Watson Research Center, Yorktown Heights, NY 10598. ATT cij = cji > 0 is the cost of the edge connecting vertices i and j, cii = 0. Consider a random walk on the graph G, executed according to a transition probability matrix P = (pij); pij is the probability that the walk moves from vertex i to vertex j, and the walk pays a cost cij in that step. Let eij (not in general equal to eji) be the expected cost of a random walk starting at vertex i and ending at vertex j (eii is the expected cost of a round trip from i). We say that the random walk has stretch c if there exists a constant a such that, for any sequence i0, i1, . . . , il of vertices ∑l j=1 eij−1ij ≤ c · ∑l j=1 cij−1ij + a. We prove the following tight result: Any random walk on a weighted graph with n vertices has stretch at least n − 1, and every weighted graph with n vertices has a random walk with stretch n− 1. The upper bound proof is constructive, and shows how to compute the transition probability matrix P from the cost matrix C = (cij). The proof uses new connections between random walks and effective resistances in networks of resistors, together with results from electric network theory. Consider a network of resistors with n vertices, and conductance σij between vertices i and j (vertices i and j are connected by a resistor with branch resistance 1/σij). Let Rij be the effective resistance between vertices i and j (i.e., 1/Rij is the current that would flow from i to j if one volt were applied between i and j; it is known that 1/Rij ≥ σij). Let the resistive random walk be defined by the probabilities pij = σij/ ∑ k σik. In Section 3 we show that this random walk has stretch n − 1 in the graph with costs cij = Rij. Thus, a random walk with optimal stretch is obtained by computing the resistive inverse (σij) of the cost matrix (cij): a network of branch conductances (σij ≥ 0), so that, for any i, j, cij is the effective (not branch)

Visual function before and after the removal of bilateral congenital cataracts in adulthood

Subject Peter Doyle (PD) had congenital bilateral cataracts removed at the age of 43. Pre-operatively PDs visual acuity was 20/80, with a resolution limit around 15 cpd, and he experienced monocular diplopia with high contrast stimuli. Post-operatively PDs visual acuity improved to approximately 20/40, with a resolution limit around 25 cpd. Using a variety of pre- and post-operative tests we have documented a wide range of neural adaptations to his limited and distorted visual input, and have found a limited amount of post-operative adaptation to his newly improved visual input. These results show that the human visual system is capable of significant adaptation to the particular optical input that is experienced.

On the bass note of a Schottky group

Using a classical method from physics called Rayleigh’s cutting method, we prove the conjecture of Phillips and Sarnak that there is a universal lower bound L2 > 0 for the lowest eigenvalue of the quotient manifold of a classical Schottky group Γ acting on hyperbolic 3-space H 3. By work of Patterson and Sullivan, this implies that there is a universal upper bound U2 < 2 for the Hausdorff dimension of the limit set of Γ, or equivalently, for the critical exponent of the Poincaré series associated with Γ. The latter implication answers a question that can be traced back to Schottky and Burnside.

Vision: Realignment of cones after cataract removal

Through unique observations of an adult case of bilateral congenital cataract removal, we have found evidence that retinal photoreceptors will swiftly realign towards the brightest regions in the pupils of the eye. Cones may be phototropic, actively orientating themselves towards light like sunflowers in a field.

Self-packing of centrally symmetric convex bodies in R 2

AbstractLetB be a compact convex body symmetric around0 in ℝ2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thattB can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaam% aalaaabaGaamyBaaqaaiabes7aKjaacIcaieqacaWFcbGaaiykaaaa% aiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaaGymaaqaaiaaik% daaaaaaOGaeyOeI0YaaSaaaeaacaaIZaaabaGaaGOmaaaacqGHKjYO% cqaHbpGCcaGGOaGaamyBaiaacYcacaWFcbGaaiykaiabgsMiJoaabm% aabaWaaSaaaeaacaWGTbaabaGaeqiTdqMaaiikaiaa-jeacaGGPaaa% aaGaayjkaiaawMcaamaaCaaaleqabaWaaSGbaeaacaaIXaaabaGaaG% OmaaaaaaGccqGHRaWkcaaIXaGaaiilaaaa!576F!

Non-sexist solution of the menage problem

\left( {\frac{m}{{\delta (B)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - \frac{3}{2} \leqslant \rho (m,B) \leqslant \left( {\frac{m}{{\delta (B)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 1,