We model bipartisan elections where voters are exposed to two forces: local homophilic interactions and external influence from two political campaigns.

The model is mathematically equivalent to the random field Ising model with a bimodal field.

When both parties exceed a critical campaign spending, the system undergoes a phase transition to a highly polarized state where homophilic influence becomes negligible, and election outcomes mirror the proportion of voters aligned with each campaign, independent of total spending.

The model predicts a hysteresis region, where the election results are not determined by campaign spending but by incumbency. Calibrating the model with historical data from US House elections between 1980 and 2020, we find the critical campaign spending to be ∼1.8 million USD.

Campaigns exceeding critical expenditures increased in 2018 and 2020, suggesting a boost in political polarization.

J. Korbel, R. Dahdoul, S. Thurner, Empirical Validation of the Polarization Transition in a Double-Random Field Model of Elections, Physical Review Letters 136 (2026) 127402.