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Through the Monte Carlo simulation of the three-dimensional, three-statePotts model, which is a paradigm of finite-temperature pure gauge QCD, we studythe fluctuations of generalized susceptibilities near the temperatures ofexternal fields of first-, second-order phase transitions and crossover.Similar peak-like fluctuation appears in the second order susceptibility atthree given external fields. Oscillation-like fluctuation appears in the thirdand fourth order susceptibilities. We find that these non-monotonicfluctuations are not only associated with the second-order phase transition,but also the first-order one and crossover in a system of finite-size. Wefurther present the finite-size scaling analysis of the second and fourth ordersusceptibilities, respectively. The exponent of the scaling characterizes theorder of the transitions, or the crossover.

Generalized susceptibilities along the phase boundary of the three-dimensional, three-state Potts model