Violations of Hyperscaling in Phase Transitions and Critical Phenomena—in Memory of Prof. Ralph Kenna

Dear Colleagues,

Professor Ralph Kenna (born 27 August 1964 – died 26 October 2023) was born in Athlone (Ireland). He was a theoretical physicist and had very diverse centres of interest such as statistical physics, complex systems and Irish mythology. After a B.A. in Theoretical Physics (1985) and an M.Sc. (1988) from Trinity College Dublin, he completed his PhD at the University of Graz under Professor Christian Lang in 1993. Kenna then moved to the University of Liverpool from 1994 to 1997 and to Trinity College Dublin from 1997 to 1999. In 2002, he was hired at Coventry University where he founded the Applied Mathematics Research Centre (joining, in 2018, the Centre for Fluid and Complex Systems. In 2016, he co-founded the L4 Collaboration and Doctoral College for the Statistical Physics of Complex Systems, joining the Universities of Coventry, Leipzig, Lorraine and the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine in Lviv (ICMP). He was a Fellow of the Institute of Mathematics and its Applications, Member of the Institute of Physics, and an Advisory Board Member of the Middle European Cooperation on Statistical Physics. In 2019, for his important scientific contributions as well as for his initiative in different forms of collaboration with Ukraine and his engagement in the preparation of young scientists, he was conferred the title of Doctor Honoris Causa of the ICMP.

In statistical physics, Kenna was a renowned expert in the study of critical phenomena and phase transitions via the analysis of the zeros of the partition function, an approach introduced by Lee and Yang in the 1950s. With his collaborators (Des Johnston and Wolfhard Janke), Kenna was noted for the development of scaling relations for logarithmic corrections, in particular at the upper critical dimension. With Bertrand Berche, he extended this to higher dimensions in 2012, following the pioneering work of Michael Fisher on dangerous irrelevant variables. This led to the introduction of the new pseudo-critical exponent ϙ (koppa) and its logarithmic counterpart ϙ (ϙ-hat) to govern the finite-size scaling (FSS) of the correlation length and a new form for FSS, called QFSS, to replace standard prescription above the upper critical dimension. Spin systems on scale-free networks also display logarithmic corrections that obey the scaling relations developed by Kenna.

Ralph Kenna has always been attracted by multidisciplinary subjects and has not hesitated during his career to tackle societal or human sciences problems. In 2010, Kenna and Berche quantified the notion of critical mass of academic research groups. Using data from the UK’s Research Assessment Exercise 2008 and the French counterpart (AERES), they tracked how research group quality depends on the size of the group. They found that quality rises linearly with group size up to a point which they later identified as akin to the Dunbar number in anthropology. Subsequently, with Olesya Mryglod and Yurij Holovatch, Kenna and Berche used scientometrics to predict the outcome of the UK’s Research Excellence Framework 2014. They found that correlations between metrics and peer review are poor.

In comparative mythology, Kenna is noted for pioneering the usage of complex networks in the study of Irish and other mythologies. His first paper on the topic was downloaded over 30,000 times in 10 years and resulted in considerable media coverage in the international press. Other major works include investigations into the Sagas of Icelanders. Kenna’s team found that whether the sagas are historically accurate or not, the properties of the social worlds they record are similar to those of real social networks. The Viking Age in Ireland as portrayed in Cogadh Gaedhel re Gallaibh was next tackled by Kenna’s team. They developed a measure to place hostility on a spectrum between civil war and international conflict. Their findings quantified and supported the traditional view of the Viking age in Ireland as one of international conflict and challenged recent revisionist claims. A study of the character Fraoch identified quantifiable differences between the two parts of his story, supporting the suggestion it was set to writing by two different scribes, one of whom embellished the tale. Kenna and co-workers also studied Ukrainian mythology. They compared the Kyiv bylyny cycle to other prominent European epics.

An imaginative and enthusiastic researcher, Ralph Kenna was a source of inspiration for his collaborators and his students.

This special issue was initiated by Prof. Ralph Kenna, then the Entropy Editorial Board Member. The topic chosen has been at the center of his activities and we owe a lot to him for his important contributions in this field. Below is a brief description of the special issue scope written by Prof. Kenna and used by him to explain the scope of the issue inviting future contributors. The Editorial Board of the Entropy journal suggested we finalize his work. We will try to do our best, in Prof. Kenna's memory. And we dedicate this Special Issue to his memory.

Universality is an emergent phenomenon, at least partially explained by the renormalization group. Because of universality, simplified theoretical models can deliver critical behaviour of real complex systems by trimming back to essentials such as dimensionality, symmetry group, and range of interaction. Universality classes of theoretical models and real systems are characterised by critical exponents that are linked through scaling relations between them. The scaling relations that involve dimensionality are referred to as hyperscaling. Due to the success of mean-field theory in highly connected systems, irrespective of the dimensionality of the systems, dimension-dependent hyperscaling is often said to fail there. That tenet was challenged recently with the introduction of new insights to the renormalization group aimed to rescue hyperscaling in high dimensions.

This Special Issue focuses on high-dimensional and other highly connected systems where hyperscaling is traditionally said to fail. We are interested in robustly supported explorations of how hyperscaling can or cannot be re-instated. The intent of this Special Issue is to capture “state of the art’’ research in high dimensions and high connectivity and as such we welcome new results and reviews of the highest standard. We are also interested in interdisciplinary applications.

Prof. Bertrand Berche
Prof. Yurij Holovatch
Guest Editors

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• hyperscaling
• finite-size scaling
• high dimensions
• mean-field
• Gaussian fixed point
• Ginzburg criterion
• dangerous irrelevancy
• critical exponents
• coppa (ϙ)
• corrections to scaling
• Fourier modes
• boundary conditions