More Like this

1. Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean saddles from Lorentzian path integrals

   https://doi.org/10.1007/JHEP07(2022)108  

   Marolf, Donald ( July 2022 , Journal of High Energy Physics)

   A bstract Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimension-2 singularities analogous to the familiar Euclidean conical singularities. With this understanding, we show that the usual Euclidean-signature black holes (or their complex rotating analogues) define saddle points for the real-time path integrals that compute our partition functions. Furthermore, when the black holes have positive specific heat, we provide evidence that a codimension-2 subcontour of our real Lorentz-signature contour of integration can be deformed so as to show that these black holes saddles contribute with non-zero weight to the semiclassical limit, and that the same is then true of the remaining two integrals. 

   more » « less
2. Minimum Weight Euclidean (1+ε)-Spanners

   Toth, Csaba D. ( October 2022 , Graph-Theoretic Concepts in Computer Science)

   Given a set S of n points in the plane and a parameter ε>0, a Euclidean (1+ε) -spanner is a geometric graph G=(S,E) that contains a path of weight at most (1+ε)∥pq∥2 for all p,q∈S . We show that the minimum weight of a Euclidean (1+ε)-spanner for n points in the unit square [0,1]2 is O(ε−3/2n−−√), and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline O(ε−2n−−√), obtained by combining a tight bound for the weight of an MST and a tight bound for the lightness of Euclidean (1+ε)-spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to d-space for all d∈N : The minimum weight of a Euclidean (1+ε)-spanner for n points in the unit cube [0,1]d is Od(ε(1−d2)/dn(d−1)/d), and this bound is the best possible. For the n×n section of the integer lattice, we show that the minimum weight of a Euclidean (1+ε)-spanner is between Ω(ε−3/4n2) and O(ε−1log(ε−1)n2). These bounds become Ω(ε−3/4n−−√) and O(ε−1log(ε−1)n−−√) when scaled to a grid of n points in [0,1]2. . 

   more » « less
3. Geoclidean: Few-Shot Generalization in Euclidean Geometry

   Joy Hsu ; Jiajun Wu ; Noah D. Goodman ( January 2023 , NeurIPS 2022 Datasets and Benchmarks)

   Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to perceive and reason with them. Will computer vision models trained on natural images show the same sensitivity to Euclidean geometry? Here we explore these questions by studying few-shot generalization in the universe of Euclidean geometry constructions. We introduce Geoclidean, a domain-specific language for Euclidean geometry, and use it to generate two datasets of geometric concept learning tasks for benchmarking generalization judgements of humans and machines. We find that humans are indeed sensitive to Euclidean geometry and generalize strongly from a few visual examples of a geometric concept. In contrast, low-level and high-level visual features from standard computer vision models pretrained on natural images do not support correct generalization. Thus Geoclidean represents a novel few-shot generalization benchmark for geometric concept learning, where the performance of humans and of AI models diverge. The Geoclidean framework and dataset are publicly available for download. 

   more » « less
4. On Euclidean Steiner (1+ε)-Spanners

   https://doi.org/10.4230/LIPIcs.STACS.2021.13  

   Bhore, Sujoy ; Toth, Csaba D. ( March 2021 , 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021))

   null (Ed.)

   Lightness and sparsity are two natural parameters for Euclidean (1+ε)-spanners. Classical results show that, when the dimension d ∈ ℕ and ε > 0 are constant, every set S of n points in d-space admits an (1+ε)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ε > 0 for constant d ∈ ℕ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+ε)-spanner. They gave upper bounds of Õ(ε^{-(d+1)/2}) for the minimum lightness in dimensions d ≥ 3, and Õ(ε^{-(d-1))/2}) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+ε)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ε^{-d/2}) for the lightness and Ω(ε^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+ε)-spanners of lightness O(ε^{-1}log n) for n points in Euclidean plane. 

   more » « less
5. The canonical ensemble reloaded: the complex-stability of Euclidean quantum gravity for black holes in a box

   https://doi.org/10.1007/JHEP08(2022)215  

   Marolf, Donald ; Santos, Jorge E. ( August 2022 , Journal of High Energy Physics)

   A bstract We revisit the stability of black hole saddles for the Euclidean path integral describing the canonical partition function Z ( β ) for gravity inside a spherical reflecting cavity. The boundary condition at the cavity wall couples the transverse-traceless (TT) and pure-trace modes that are traditionally used to describe fluctuations about Euclidean Schwarzschild black holes in infinite-volume asymptotically flat and asymototically AdS spacetimes. This coupling obstructs the familiar Gibbons-Hawking-Perry treatment of the conformal factor problem, as Wick rotation of the pure-trace modes would require that the TT modes be rotated as well. The coupling also leads to complex eigenvalues for the Lichnerowicz operator. We nevertheless find that the Lichnerowicz operator can be diagonalized in the space of coupled modes. This observation allows the eigenmodes to define a natural generalization of the pure-trace Wick-rotation recipe used in infinite volume, with the result that a mode with eigenvalue λ is stable when Re λ > 0. In any cavity, and with any cosmological constant Λ ≤ 0, we show this recipe to reproduce the expectation from black hole thermodynamics that large Euclidean black holes define stable saddles while the saddles defined by small Euclidean black holes are unstable. 

   more » « less