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The critical point is a fixed point in finite-size scaling. To quantify thebehaviour of such a fixed point, we define, at a given temperature and scalingexponent ratio, the width of scaled observables for different sizes. Theminimum of the width reveals the position of fixed point, its correspondingphase transition temperature, and scaling exponent ratio. The value of thisratio tells the nature of fixed point, which can be a critical point, a pointof the first order phase transition line, or a point of the crossover region.To demonstrate the effectiveness of this method, we apply it to three typicalsamples produced by the 3D three-state Potts model. Results show the method tobe more precise and effective than conventional methods. Finally, we discuss apossible application at the beam energy scan program of relativistic heavy-Ioncollision.

Locating fixed points in the phase plane