Paper Citation Record · LEDGER

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$

As of 13 August 2026, this Paper Citation Record lists 33 of 33 outbound references and 0 inbound Pith citation observations for arXiv:2607.10510.

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pith.paper-citation-record.v1
2607.10510 v1

Coverage vector

measured 33 of 33 reference resolution

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Source: paper_references, paper_reference_links, observed 2026-07-14T11:12:38.453104Z

measured 33 of 33 standing notices

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Source: scholarly_work_events, retraction_status_cache, observed 2026-08-13T06:32:02.005865+00:00

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Reference resolution

33 of 33 outbound references displayed

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Outbound references

Observation 61d3fce3-cf25-4453-ab44-917194a2d327 · outbound

This paper cites A cluster expansion approach to the Heilmann–Lieb liquid crystal model.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ A cluster expansion approach to the Heilmann–Lieb liquid crystal model

Reference 1

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Observation 1418f65b-9d99-4c13-afe3-ba955b32310f · outbound

This paper cites an unresolved cited work.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Unresolved cited work

Reference 2

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Observation 6498eed4-99a1-4205-8272-caa7987f4b0f · outbound

This paper cites Entropy of fully packed hard rigid rods ond-dimensional hyper- cubic lattices.Physical Review E, 103(4):042130, 2021.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Entropy of fully packed hard rigid rods ond-dimensional hyper- cubic lattices.Physical Review E, 103(4):042130, 2021

Reference 3

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Observation 19efb8c1-e514-4cbf-815c-5eb50cd47602 · outbound

This paper cites The nematic phase of a system of long hard rods.Communications in Mathematical Physics, 323(1):143–175, 2013.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ The nematic phase of a system of long hard rods.Communications in Mathematical Physics, 323(1):143–175, 2013

Reference 4

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Observation 4783ea67-17fb-46b9-92f6-2d8226a4691f · outbound

This paper cites Plate-nematic phase in three dimensions.Communications in Mathematical Physics, 373(1):327–356, 2020.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Plate-nematic phase in three dimensions.Communications in Mathematical Physics, 373(1):327–356, 2020

Reference 5

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Observation 949b1067-e9de-4a70-b1cb-5b89db427104 · outbound

This paper cites Onsager’s spherocylinders revisited.Journal of Physical Chemistry, 91(19):4912–4916, 1987.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Onsager’s spherocylinders revisited.Journal of Physical Chemistry, 91(19):4912–4916, 1987

Reference 6

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Observation 3b9712c7-4b30-478f-9ee6-fcd19d9c5e09 · outbound

This paper cites Cambridge University Press, 2017.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Cambridge University Press, 2017

Reference 7

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Observation eae5b56b-e73a-40a7-b7fd-f798058cb150 · outbound

This paper cites On the orientational ordering of long rods on a lattice.Europhysics Letters, 78(2):20003, 2007.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ On the orientational ordering of long rods on a lattice.Europhysics Letters, 78(2):20003, 2007

Reference 8

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Observation 6a24625e-b212-43e6-b06a-70e5ef53ee7f · outbound

This paper cites Columnar order in random packings of 2×2 squares on the square lattice.Journal of the European Mathematical Society, 2025.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Columnar order in random packings of 2×2 squares on the square lattice.Journal of the European Mathematical Society, 2025

Reference 9

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Observation 3c51f115-890c-4c52-b740-d60c610474df · outbound

This paper cites Extended regime of nematic order in an interacting monomer-dimer model of Heilmann and Lieb.arXiv preprint arXiv:2511.23437, 2025.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Extended regime of nematic order in an interacting monomer-dimer model of Heilmann and Lieb.arXiv preprint arXiv:2511.23437, 2025

Reference 10

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Observation 4f82b43d-4cd9-4327-850e-8d3f6308efc2 · outbound

This paper cites Theory of monomer-dimer systems.Communications in Mathematical Physics, 25(3):190–232, 1972.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Theory of monomer-dimer systems.Communications in Mathematical Physics, 25(3):190–232, 1972

Reference 11

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Observation 04f62a82-e8d8-4537-876f-3c071bdff1e5 · outbound

This paper cites Lattice models for liquid crystals.Journal of Statistical Physics, 20(6):679–693, 1979.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Lattice models for liquid crystals.Journal of Statistical Physics, 20(6):679–693, 1979

Reference 12

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Observation 89c889bc-69d0-491a-afdb-d948ab33ba8b · outbound

This paper cites Nematic liquid crystal phase in a system of interacting dimers and monomers.Communications in Mathematical Physics, 363(3):955–1002, 2018.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Nematic liquid crystal phase in a system of interacting dimers and monomers.Communications in Mathematical Physics, 363(3):955–1002, 2018

Reference 13

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Observation 86bc19ed-fd95-465c-9437-cce3b76399b3 · outbound

This paper cites Phase transitions in a system of hard rectangles on the square lattice.Physical Review E, 89(5):052124, 2014.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Phase transitions in a system of hard rectangles on the square lattice.Physical Review E, 89(5):052124, 2014

Reference 14

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Observation cbdcfa83-015c-423a-958b-2f5c515c85b7 · outbound

This paper cites Asymptotic behavior of the isotropic-nematic and nematic- columnar phase boundaries for the system of hard rectangles on a square lattice.Physical Review E, 91(1):012105, 2015.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Asymptotic behavior of the isotropic-nematic and nematic- columnar phase boundaries for the system of hard rectangles on a square lattice.Physical Review E, 91(1):012105, 2015

Reference 15

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Observation fb04153f-7c9e-4640-a14f-a7ab55bb6b78 · outbound

This paper cites Phase transitions in systems of hard rectangles with non-integer aspect ratio.The European Physical Journal B, 88(5):133, 2015.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Phase transitions in systems of hard rectangles with non-integer aspect ratio.The European Physical Journal B, 88(5):133, 2015

Reference 16

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Observation f43198b5-7e40-4bd7-8dc2-64f0a8849b56 · outbound

This paper cites an unresolved cited work.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Unresolved cited work

Reference 17

Resolution: unresolved; no resolver link, observed 2026-07-14T11:12:38.453104Z

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Observation 53617683-9d8b-47c6-8a49-7a52d8bf4687 · outbound

This paper cites Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations.The Journal of Chemical Physics, 128(21), 2008.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations.The Journal of Chemical Physics, 128(21), 2008

Reference 18

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Observation ab177ed2-9e97-48d8-869a-f2db0b13ce66 · outbound

This paper cites High-density hard-core model on triangular and hexagonal lattices.Annales Henri Poincar´ e, 26(9):3321–3381, 2025.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ High-density hard-core model on triangular and hexagonal lattices.Annales Henri Poincar´ e, 26(9):3321–3381, 2025

Reference 19

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Observation 883bbf03-fb63-4da1-aa74-8917be643918 · outbound

This paper cites Stability of columnar order in assemblies of hard rectangles or squares.Europhysics Letters, 114(1):10003, 2016.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Stability of columnar order in assemblies of hard rectangles or squares.Europhysics Letters, 114(1):10003, 2016

Reference 20

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Observation 0d69f6fd-5bea-411e-8481-544680e5a7af · outbound

This paper cites High-activity expansion for the columnar phase of the hard rectangle gas.Journal of Statistical Physics, 160(5):1173–1197, 2015.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ High-activity expansion for the columnar phase of the hard rectangle gas.Journal of Statistical Physics, 160(5):1173–1197, 2015

Reference 21

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Observation 3c8fc401-b446-466d-943c-53c90865e3f8 · outbound

This paper cites The effects of shape on the interaction of colloidal particles.Annals of the New York Academy of Sciences, 51(4):627–659, 1949.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ The effects of shape on the interaction of colloidal particles.Annals of the New York Academy of Sciences, 51(4):627–659, 1949

Reference 22

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Observation bfe94783-adb9-4996-bac3-766c1ea4757c · outbound

This paper cites A condition for long-range order in discrete spin systems with application to the antiferromagnetic Potts model.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ A condition for long-range order in discrete spin systems with application to the antiferromagnetic Potts model

Reference 23

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Observation 5ffa932d-e181-43ac-924a-c2951c322de6 · outbound

This paper cites Long-range order in discrete spin systems.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Long-range order in discrete spin systems

Reference 24

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Observation de5fec81-acbc-4315-970e-1e58d1179452 · outbound

This paper cites Rigidity of proper colorings ofZ d.Inventiones mathematicae, 232(1):79–162, 2023.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Rigidity of proper colorings ofZ d.Inventiones mathematicae, 232(1):79–162, 2023

Reference 25

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Observation 4c3dbef2-aa34-45fe-a68f-b637f6833ba8 · outbound

This paper cites Springer, 2009.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Springer, 2009

Reference 26

Resolution: unresolved; no resolver link, observed 2026-07-14T11:12:38.453104Z

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Observation b73f7fad-cc42-4c75-abff-d64ef7d3c39e · outbound

This paper cites Phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice.Physical Review E, 105(3):034103, 2022.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice.Physical Review E, 105(3):034103, 2022

Reference 27

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Observation 84666bf9-ff8f-4372-af7c-33e0cabecd44 · outbound

This paper cites Boundary-connectivity via graph theory.Proceedings of the American Mathe- matical Society, 141(2):475–480, 2013.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Boundary-connectivity via graph theory.Proceedings of the American Mathe- matical Society, 141(2):475–480, 2013

Reference 28

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Observation 7e9f7a4c-3365-46a9-b8e6-0bb51d92eb7b · outbound

This paper cites Cluster expansions and correlation functions.Moscow Mathematical Journal, 4(2):511–522, 2004.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Cluster expansions and correlation functions.Moscow Mathematical Journal, 4(2):511–522, 2004

Reference 29

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Observation a33622ee-18c5-4ecf-b813-66d70d8f44ce · outbound

This paper cites Generalized Onsager theory of liquid crystals.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 88(6):062501, 2013.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ Generalized Onsager theory of liquid crystals.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 88(6):062501, 2013

Reference 30

Resolution: unresolved; no resolver link, observed 2026-07-14T11:12:38.453104Z

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Observation c1b1762b-a0ca-4895-bab7-7e27e67af37c · outbound

This paper cites An alternate version of Pirogov-Sinai theory.Communications in Mathe- matical Physics, 93(4):559–581, 1984.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ An alternate version of Pirogov-Sinai theory.Communications in Mathe- matical Physics, 93(4):559–581, 1984

Reference 31

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Observation b611c52d-5c56-4d4e-b1f7-af11f440cff7 · outbound

This paper cites 10(ii) θ3 (0, 1.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ 10(ii) θ3 (0, 1

Reference 32

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Observation c301513b-832b-443c-8a15-626eeb57f3d6 · outbound

This paper cites 10(iii) θ4 (0,1) Lem.

Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\mathbb{Z}^2$ 10(iii) θ4 (0,1) Lem

Reference 33

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