ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P4.05
https://doi.org/10.26493/1855-3974.3114.d47
(Also available at http://amc-journal.eu)
Complexity function of jammed configurations
of Rydberg atoms*
Tomislav Došlić
Department of Mathematics, Faculty of Civil Engineering,
University of Zagreb, Zagreb, Croatia and
Faculty of Information Studies, Novo Mesto, Slovenia
Mate Puljiz , Stjepan Šebek † , Josip Žubrinić
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing,
University of Zagreb, Zagreb, Croatia
Received 3 May 2023, accepted 4 March 2024, published online 25 September 2024
Abstract
In this article, we determine the complexity function (configurational entropy) of
jammed configurations of Rydberg atoms on a one-dimensional lattice. Our method con-
sists of providing asymptotics for the number of jammed configurations determined by
direct combinatorial reasoning. In this way we reduce the computation of complexity to
solving a constrained optimization problem for the Shannon’s entropy function. We show
that the complexity can be expressed explicitly in terms of the root of a certain polynomial
of degree b, where b is the so-called blockade range of a Rydberg atom. Our results are
put in a relation with the model of irreversible deposition of k-mers on a one-dimensional
lattice.
Keywords: Dynamic lattice systems, equilibrium lattice systems, complexity function, configurational
entropy, jammed configuration, maximal packing, Rydberg atoms.
Math. Subj. Class. (2020): 82B20, 82C20, 05B40, 05A15, 05A16
*It is a pleasure for us to thank Jean-Marc Luck and Pavel Krapivsky for fruitful exchanges during the concomi-
tant elaboration of their preprint [43] and of the present work. T. Došlić gratefully acknowledges partial support
by the Slovenian ARIS via Program P1-0383, grant no. J1-3002, and by COST Action CA21126 NanoSpace.
Financial support of the Croatian Science Foundation (project IP-2022-10-2277) is gratefully acknowledged by
S. Šebek.
Lastly, we thank the anonymous referee for helpful comments that have led to improvements of the presentation
of the article.
†Corresponding author.
E-mail addresses: tomislav.doslic@grad.unizg.hr (Tomislav Došlić), mate.puljiz@fer.unizg.hr (Mate
Puljiz), stjepan.sebek@fer.unizg.hr (Stjepan Šebek), josip.zubrinic@fer.unizg.hr (Josip Žubrinić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P4.05
https://doi.org/10.26493/1855-3974.3114.d47
(Dostopno tudi na http://amc-journal.eu)
Kompleksnostna funkcija zataknjenih
konfiguracij Rydbergovih atomov*
Tomislav Došlić
Department of Mathematics, Faculty of Civil Engineering,
University of Zagreb, Zagreb, Croatia and
Faculty of Information Studies, Novo Mesto, Slovenia
Mate Puljiz , Stjepan Šebek † , Josip Žubrinić
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing,
University of Zagreb, Zagreb, Croatia
Prejeto 3. maja 2023, sprejeto 4. marca 2024, objavljeno na spletu 25. septembra 2024
Povzetek
V tem članku določimo kompleksnosto funkcijo (konfiguracijsko entropijo) zatakn-
jenih konfiguracij Rydbergovih atomov na enodimenzionalni mreži. Naša metoda sestoji
iz določitve asimptotike za število zataknjenih konfiguracij, ki jo določimo z neposrednim
kombinatoričnim sklepanjem. Na ta način reduciramo računanje kompleksnosti na reše-
vanje omejenega optimizacijskega problema za Shannonovo entropijsko funkcijo. Pokaže-
mo, da se da kompleksnost izraziti eksplicitno v smislu ničel določenega polinoma stopnje
b, kjer je b t.i. razpon blokade Rydbergovega atoma. Naši rezultati so postavljeni v zvezo z
modelom nereverzibilnega odlaganja k-merov na enodimenzionalni mreži.
Ključne besede: Dinamični mrežni sistemi, ravnovesni mrežni sistemi, kompleksnostna funkcija, kon-
figuracijska entropija, zataknjena konfiguracija, maksimalno pakiranje, Rydbergovi atomi.
Math. Subj. Class. (2020): 82B20, 82C20, 05B40, 05A15, 05A16
*Z veseljem se zahvaljujemo Jean-Marc Lucku in Pavlu Krapivskemu za plodno izmenjavo med sočasno
pripravo njunega predtiska [43] in tega dela. T. Došlić se zahvaljuje za delno podporo Javne agencije
za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije ARIS v okviru programa P1-0383,
nepovratna sredstva št. J1-3002, in s strani COST Action CA21126 NanoSpace. S. Šebek. se zahvaljuje za
finančno podporo Hrvaške znanstvene fundacije (projekt IP-2022-10-2277).
Nazadnje se zahvaljujemo neznanemu recenzentu za koristne pripombe, ki so vodile k izboljšavi razumljivosti
članka.
†Kontaktni avtor.
E-poštni naslovi: tomislav.doslic@grad.unizg.hr (Tomislav Došlić), mate.puljiz@fer.unizg.hr (Mate Puljiz),
stjepan.sebek@fer.unizg.hr (Stjepan Šebek), josip.zubrinic@fer.unizg.hr (Josip Žubrinić)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/