We investigate two randomly frusuated ID or quasi-ID Ising systems with diluted disorder. namely the ferromagnetic chain in a random field and the ladder spin glass. We present an analytical study of the susceptibilities (linear x . nonlinear x g , and higher orders), which characterize the response to a uniform external field at finite temperature. Both models admit a continuum description in the scaling regime of low tempemre and low impurity concentration, where the susceptibilities obey power laws, similarly to usual crihcal phenomena, albeit unlike the man-field theory of spin glasses. We obtain explicit expressions for the scaling functions of x and x 3 , and an estimate for the essential Lee-Yang singuluty. $ 4379 4380The coefficients xZ-, are the susceptibilities of the model. The linear suscepribility x , which characterizes the linear response of the system, can be expressed as the following sum of the connected two-point correlation function in zero field: E Amic nnd J M Luck whereas the nonlinenr susceptibility x3 can be expressed as a sum of the four-point correlations of the spins, and so on for the higher-order susceptibilities xu-, .Exactly solvable one-dimensional (ID) disordered magnetic systems provide interesting test cases. In spite of the absence of a finite-temperature phase transition, frustration is responsible for the existence of numerous almost-degenerate states, yielding a rich low-temperature behaviour in thermodynamical quantities and correlation functions. The ferromagnetic king chain in a quenched random magnetic field is certainly the simplest of these model systems. It has been the subject of many investigations, based on the observation that its free energy is the Lyapunov exponent of an infinite product of random 2 x 2 transfer matrices (see [4,5] for reviews). Its thermodynamical properties at finite temperature are known exactly only for some classes of distributions of the quenched random fields, either discrete [6] or continuous [7-91. Some results are also available for more general distributions of disorder, either at zero temperature [lo], or in the weakdisorder regime [11][12][13]. The two-point correlation function and the linear susceptibility x are only known in a limited number of cases [4,6,14], whereas virtually nothing is known about higher-order correlations and susceptibilities. The converse situation of an king chain with random exchange couplings is not frustrated, but the ladder geometry, i.e. two chains coupled by transversal exchange couplings, is frustrated, and thus provides an interesting spin-glass model [IO. 151.The purpose of the present article is a detailed study of the higher-order susceptibilities of ID disordered magnetic models, and particularly a comparison of their scaling behaviour with the cases of common critical phenomena and of the mean-field theory of spin glasses. The analytical investigation of the susceptibilities is performed on the ID disordered king models mentioned above, namely the ferromagnetic chain in a random magnetic fiel...