A Generalisation of the Ising Model on the Complete Graph

We investigate a generalisation of the Ising model on the complete graph. The Ising model on the complete graph is known as the Curie-Weiss model, and its Hamiltonian can be generalised for higher-dimensional spins by replacing the product of spins with a dot product. We use this Hamiltonian to investigate a model that has not previously been studied, where spins are vectors of length n containing elements in {-1,0,1}, with precisely k non-zero elements for a fixed k. Under this model, we will calculate the moment generating function of the sum of spins, also known as the magnetisation. We will use this result to determine central limit theorems and investigate their dependence on temperature for given k and n. Finally, we aim to deduce the existence of a phase transition and its dependence on k and n.

Emily Palit

Monash University

Emily Palit is in her second year of a Bachelor of Science Advanced – Research (Honours) at Monash University, with an extended major in mathematics. Emily has strong interests in topics including probability theory, statistics and optimisation. By completing an AMSI summer project, Emily is hoping to gain research experience and consolidate her interests for further studies in mathematics.

While mathematics is her primary interest, Emily has excelled in other fields of science such as chemistry and computational science. She has gained accolades including the Dean’s List Award (2022) and the Jackson Prize for Chemistry I Advanced (2022). Last summer, Emily completed a data analysis research project studying the relationship between blocking high pressure systems and flooding. She shared her findings at the Weather and Climate Interactions Workshop in February, 2023. Outside of her own studies, Emily strengthens her skills in clear communication by tutoring high school students in mathematics. When she is not doing maths, she loves to get outdoors and hike up mountains.

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