Rogue waves are rapid and unpredictable events of exceptional amplitude reported in various fields, such as oceanography and optics, with much of the interest being targeted towards their physical origins and likelihood of occurrence. Here, we use the all-round framework of discrete-time quantum walks to study the onset of those events due to a random phase modulation, unveiling its long-tailed statistics, survival time, and dependence upon the degree of randomness. We find the minimal disorder strength allowing for their occurrence to scale $\propto N^{-1/2}, N$ being the number of sites. Moreover, an extreme-value analysis converges to the Gumbel class of limiting distributions.

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• Anderson localization
• Quantum walks
• Extreme event statistics